Functions of a Complex Variable and Limits

A function of a complex variable looks, at first glance, like nothing new: a rule $f$ sending each $z$ of a domain $\Omega\subseteq\mathbb{C}$ to a value $w=f(z)$ , exactly as a real function sends a number to a number. But writing $z=x+iy$ and $f(z)=u(x,y)+i\,v(x,y)$ exposes a double life. The single complex object $f$ is simultaneously a pair of real scalar fields $u,v$ over the plane, and every question about $f$ can be asked twice — once in one complex variable, once in two real variables. Switching fluently between those dialects is the grammar of complex analysis. The pivot on which the subject turns is the humble notion of a limit. Because $\mathbb{C}$ under the modulus distance $|z_1-z_2|$ is nothing but the Euclidean plane $\mathbb{R}^2$ with a metric, the $\varepsilon$ – $\delta$ definition copies over verbatim from real analysis — yet it means something far stronger. On the real line a point has only a left and a right neighbour, so $x\to x_0$ is a two-sided squeeze. In the plane $z_0$ is surrounded: an approaching point can spiral, oscillate, or arrive along any curve, and $\lim_{z\to z_0}f(z)$ exists only if every one of those infinitely many approaches yields the same value. That single demand — path-independence — is the seed of everything. Sharpen the demand from the function itself to its difference quotient $\frac{f(z)-f(z_0)}{z-z_0}$ and you get complex differentiability, from which the Cauchy–Riemann equations, contour integration, the residue calculus, and Taylor and Laurent series all unfold. Limits and continuity are the quiet gateway to that entire edifice.

Key takeaways

A complex function $f:\Omega\to\mathbb{C}$ assigns to each $z=x+iy$ a value $w=u(x,y)+i\,v(x,y)$ , so it is equivalently a pair of real functions of two real variables.
Since $\mathbb{C}$ with the metric $|z_1-z_2|$ is isometric to $\mathbb{R}^2$ , the $\varepsilon$ – $\delta$ definition transfers unchanged, and $\lim_{z\to z_0}f=L$ splits into $\lim u=\operatorname{Re}L$ and $\lim v=\operatorname{Im}L$ .
The decisive new feature: $z$ may reach $z_0$ along infinitely many paths, and the limit exists only if all agree — far stronger than a one-dimensional limit.
The workhorse counterexample is $\lim_{z\to0}\bar z/z=e^{-2i\theta}$ : constant modulus $1$ but a different phase for every direction $\theta$ , so no limit exists.
Continuity is cheap; path-independence of the difference quotient is rare, and it is exactly what promotes continuity to the Cauchy–Riemann equations $u_x=v_y,\ u_y=-v_x$ .
Key idea: in the plane $z$ can approach $z_0$ along infinitely many paths, so $\lim_{z\to z_0}f(z)$ exists only when every path yields the same value — this path-independence is what separates complex analysis from two-variable real calculus and is the germ of complex differentiability.

The core insight

In the plane $z$ can approach $z_0$ along infinitely many paths; $\lim_{z\to z_0}f(z)$ exists only when every path yields the same value. This path-independence is what distinguishes complex analysis from two-variable real calculus and is the seed of complex differentiability.

Where this lives today

The abstract machinery of complex limits runs quietly inside a surprising amount of working software and physics. Every time a numerical library evaluates a function of a complex argument — `numpy`'s `cmath`, a computer-algebra system, a lattice-QCD propagator — it must respect the branch cuts of $\log$ , $\sqrt{\ }$ , and $z^{a}$ , which are just the places where a limit fails to be single-valued as you circle a branch point; get the cut wrong and the phase jumps by $2\pi$ in the wrong hemisphere. Domain-coloring plots encode $\arg f(z)$ as hue and $|f(z)|$ as brightness precisely so a human eye can see where limits blow up (poles), vanish (zeros), or refuse to settle (essential singularities, which shimmer chaotically — the Casorati–Weierstrass theorem made visible). The same structure underwrites analytic continuation: the Riemann zeta function continues a divergent series past its natural boundary, and in quantum field theory dimensional regularization and the continuation of scattering amplitudes into complex energy hinge on how a function behaves as $z$ nears a singular point. Conformal maps still solve real two-dimensional problems in electrostatics, ideal fluid flow, and impedance tomography. And complex dynamics — iterating $z\mapsto z^2+c$ on the Riemann sphere — turns "where does the limit of the orbit live?" into the boundary of the Mandelbrot set, one of the most-computed objects in all of mathematics.

The physical picture

Set two copies of the plane side by side: the $z$ -plane on the left, the $w$ -plane on the right. A complex function grabs a point from the left plane and stamps down a point in the right. Continuity is the promise that the machine does not tear the fabric: wiggle the input inside a tiny disc and the output stays inside a tiny disc, stretched or shrunk but never ripped apart. The subject is the study of which stamping rules keep this promise — and, later, which keep it so smoothly they preserve angles too. The first figure makes the $\varepsilon$ – $\delta$ definition visual, with a concrete radius attached.

Figure

The ε–δ picture of continuity for f(z) = z² at z₀ = 1: every input within a disc of radius δ = ε/3 around z₀ is stamped into the disc of radius ε around L = f(z₀) = 1, so the δ-disc on the left maps inside the ε-disc on the right

Now the mental upgrade that makes the plane different from the line. A point on the real axis is approached from two directions; a point in the plane is besieged from every direction and along every curve. So "approaching $z_0$ " is not a single squeeze but a swarm of paths closing in, and a limit must return one answer to the entire swarm — a stringent vow whose reward is the rigidity that makes complex analysis so powerful. The cleanest way to feel how the vow fails is $f(z)=\bar z/z$ , whose modulus is a constant $1$ yet whose value swings around the whole unit circle as the direction of approach rotates. The second figure shows the swarm and its images.

Figure

The path-dependence test: as z → 0 along the ray z = r·e^{iθ}, the quotient z̄/z equals e^{−2iθ} and lands at a different point of the unit circle for each direction — θ = 0° gives +1, θ = 45° gives −i, θ = 90° gives −1, θ = 135° gives +i — so the modulus is always 1 but no single limit exists

Two lessons live in these pictures. First, the metric $|z_1-z_2|$ is the same Euclidean distance you trust in $\mathbb{R}^2$ , so nothing topological is new — open discs, closed sets, connectedness all carry over; what is new is that the plane also carries a multiplication, and a limit must respect both the geometry and that algebra at once. Second, continuity of $f$ and of $|f|$ are not the same promise: $\bar z/z$ has a perfectly continuous modulus and no limit at all.

The core equations

A complex function is bookkeeping for two real fields. The single most useful move in the subject is to split $f$ and any target value into real and imaginary parts, converting one complex statement into a pair of familiar two-variable real statements.

f:\Omega\to\mathbb{C},\qquad f(z)=u(x,y)+i\,v(x,y),\qquad z=x+iy \qquad (1)

where: $\Omega$ — the domain, usually an open connected subset of $\mathbb{C}$ ; $z=x+iy$ — the input, with real part $x$ and imaginary part $y$ ; $u,v$ — real-valued functions of the two real variables $x,y$ ; $w=f(z)$ — the complex output. Reading: $f$ is one object and two fields at the same time, and every theorem can be phrased in either language. The limit is defined by the identical $\varepsilon$ – $\delta$ words as in real analysis, but the modulus now measures distance in the plane, so "close" means "inside a disc" rather than "inside an interval".

\lim_{z\to z_0}f(z)=L \iff \forall\varepsilon>0\ \exists\delta>0:\ 0<|z-z_0|<\delta \Rightarrow |f(z)-L|<\varepsilon \qquad (2)

where: $z_0$ — the limit point, which need not lie in $\Omega$ but must be approachable within it; $L$ — the candidate limit in $\mathbb{C}$ ; $|z-z_0|$ — the modulus distance, identical to the Euclidean distance in $\mathbb{R}^2$ ; $\varepsilon,\delta$ — the output tolerance and the radius of the deleted input disc that meets it. Reading: the condition $0<|z-z_0|$ deletes the centre, so the value $f(z_0)$ (if any) never enters the definition of the limit. Splitting (2) into components turns one complex limit into two real ones — the bridge that lets multivariable calculus do the work.

\lim_{z\to z_0}f(z)=a+ib \iff \lim_{(x,y)\to(x_0,y_0)}u(x,y)=a \ \text{ and }\ \lim_{(x,y)\to(x_0,y_0)}v(x,y)=b \qquad (3)

where: $a=\operatorname{Re}L,\ b=\operatorname{Im}L$ — real and imaginary parts of the target; $u,v$ — the component fields of $f$ . Reading: a complex limit exists exactly when both component limits exist, so every pathology of two-variable limits reappears here. Continuity is the limit condition with the deleted centre filled back in and made to agree.

f\ \text{continuous at}\ z_0 \iff \lim_{z\to z_0}f(z)=f(z_0) \qquad (4)

where: $f(z_0)$ — the value at the point, which must now exist and equal the limit. Reading: continuity bundles three requirements — the value exists, the limit exists, and they coincide — and by (3) it holds iff both $u$ and $v$ are continuous at $(x_0,y_0)$ . The reverse triangle inequality is the quiet engine behind almost every $\varepsilon$ – $\delta$ estimate; it also shows at a stroke that continuity of $f$ forces continuity of $|f|$ .

|z_1+z_2|\le|z_1|+|z_2|,\qquad \big||z_1|-|z_2|\big|\le|z_1-z_2| \qquad (5)

where: $z_1,z_2$ — arbitrary complex numbers. Reading: the left bound controls sums, the right (reverse) form controls moduli; setting $z_1=f(z),\,z_2=f(z_0)$ gives $\big||f(z)|-|f(z_0)|\big|\le|f(z)-f(z_0)|$ , so $f\to f(z_0)$ drags $|f|\to|f(z_0)|$ — never the converse. Finally, sharpen continuity by dividing the increment of $f$ by the increment of $z$ and demanding the quotient have a path-independent limit — the entire content of complex differentiability.

f'(z_0)=\lim_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z},\qquad \Delta z\ \text{free to approach}\ 0\ \text{along any path} \qquad (6)

where: $\Delta z$ — the complex increment, free to arrive along any curve; $f'(z_0)$ — the complex derivative, which exists only if every path returns the same value. Reading: this is a two-dimensional limit in disguise, and its path-independence, examined along the $x$ - and $y$ -axes, becomes the Cauchy–Riemann equations $u_x=v_y,\ u_y=-v_x$ .

Derivation AA limit that fails — the barred-z counterexample

The goal is to run the path-dependence test in full on the single most instructive map in the theory, $f(z)=\bar z/z$ , and extract the moral that governs every limit computation to come. We approach the origin along two special directions, find two different answers, then compute the answer along an arbitrary direction to see the value spread continuously around a circle — proving not merely that the limit fails but exactly how it fails, the diagnostic skill worth internalising.

Step 1. Approach along the real axis. Put $z=x$ with $x$ real and $x\to 0$ ; then $\bar z=x$ too, so $\bar z/z=x/x=1$ . Every real point returns the value $1$ . Why: the real axis is the "obvious" path, and if you stopped here you would wrongly guess the limit is $1$ .

Step 2. Approach along the imaginary axis. Put $z=iy$ with $y$ real and $y\to 0$ ; then $\bar z=-iy$ , so $\bar z/z=-iy/iy=-1$ . The imaginary axis returns $-1\neq 1$ , and two disagreeing paths are already enough to deny the limit. Why: it shows the failure is not a fluke of one bad direction — the two most natural approaches already contradict each other.

Step 3. Approach along a general ray. Write $z=r\,e^{i\theta}$ with fixed direction $\theta$ and $r\to 0$ ; then $\bar z=r\,e^{-i\theta}$ and $\bar z/z=e^{-2i\theta}$ , independent of $r$ . As $\theta$ runs over $[0,\pi)$ the value $e^{-2i\theta}$ sweeps the entire unit circle. Why: this exhibits the full one-parameter family of "limit candidates", one per direction, so the failure is maximal — not one rogue value but a whole circle of them.

Step 4. Conclude. No single $L$ can equal $e^{-2i\theta}$ for all $\theta$ , so $\lim_{z\to0}\bar z/z$ does not exist — even though $|\bar z/z|=1$ has a perfectly good limit. The lesson: a constant modulus never rescues a limit, and agreement along the axes is necessary but nowhere near sufficient. Why: this is the template you carry to every limit — test at least a family of paths, and trust only path-independent agreement.

Go deeper · Derivation B · 1 minContinuity of z² straight from ε and δ

Derivation BContinuity of z² straight from ε and δ

The goal is to prove $f(z)=z^2$ continuous at an arbitrary $z_0\in\mathbb{C}$ using nothing but the definition, and in doing so to expose the universal three-move template — factor the difference, bound the runaway factor on a scratch disc, then choose $\delta$ — that proves continuity of every polynomial. The concrete $\delta=\varepsilon/3$ at $z_0=1$ is the number annotated in the first figure, so this derivation is where that picture comes from.

Step 1. Factor the difference exactly. Over $\mathbb{C}$ we have the algebraic identity $|z^2-z_0^2|=|z-z_0|\,|z+z_0|$ . One factor, $|z-z_0|$ , is the quantity we control; the other, $|z+z_0|$ , is a nuisance that grows with $z$ and must be tamed. Why: isolating the controllable factor from the nuisance factor is the move that makes the estimate tractable.

Step 2. Bound the nuisance factor on a scratch disc. Provisionally restrict attention to $|z-z_0|<1$ . Then, writing $z+z_0=(z-z_0)+2z_0$ and using the triangle inequality, $|z+z_0|\le|z-z_0|+2|z_0|<1+2|z_0|$ . So on this disc the nuisance factor is bounded by the constant $1+2|z_0|$ . Why: without a preliminary cap the product could blow up; pinning $z$ to a unit disc converts the nuisance factor into a fixed number.

Step 3. Choose $\delta$ to beat $\varepsilon$ . Take $\delta=\min\!\big(1,\ \varepsilon/(1+2|z_0|)\big)$ . The first entry keeps us on the scratch disc so Step 2 stays valid; the second forces $|z-z_0|\,(1+2|z_0|)<\varepsilon$ . Then $|z^2-z_0^2|=|z-z_0|\,|z+z_0|<\delta\,(1+2|z_0|)\le\varepsilon$ . Why: the $\min$ is the standard trick for honouring two competing constraints — stay local, and stay under tolerance — with a single radius.

Step 4. Read off the number and generalise. At $z_0=1$ this gives $\delta=\min(1,\varepsilon/3)=\varepsilon/3$ for small $\varepsilon$ , the value drawn in the figure. The very same factor–bound–choose template, applied term by term, proves every polynomial continuous on all of $\mathbb{C}$ , and every rational function continuous wherever its denominator is nonzero. Why: one clean proof, reused, dispatches an entire class of functions — the economy that makes $\varepsilon$ – $\delta$ worth the effort.

Go deeper · Derivation C · 1 minThe difference quotient separates z² from barred-z

Derivation CThe difference quotient separates z² from barred-z

The goal is to show by direct computation why continuity is not enough for differentiability, and thereby to motivate the Cauchy–Riemann equations that the next article owns. Both $z^2$ and $\bar z$ are continuous everywhere; we form the difference quotient of each and let the increment approach zero along two directions, and only one survives the test — the difference is precisely path-independence, whose component form is the Cauchy–Riemann system.

Step 1. Compute the quotient for $z^2$ . Expanding, $\dfrac{(z_0+\Delta z)^2-z_0^2}{\Delta z}=\dfrac{2z_0\,\Delta z+\Delta z^2}{\Delta z}=2z_0+\Delta z$ . As $\Delta z\to 0$ this tends to $2z_0$ no matter how $\Delta z$ shrinks, because the leftover term $\Delta z$ vanishes on every path. So $\frac{\mathrm{d}}{\mathrm{d}z}z^2=2z$ , path-independent. Why: the increment cancels cleanly, leaving a direction-free limit — the signature of a genuine complex derivative.

Step 2. Compute the quotient for $\bar z$ . Here $\dfrac{\overline{z_0+\Delta z}-\bar z_0}{\Delta z}=\dfrac{\overline{\Delta z}}{\Delta z}$ , which is exactly the barred-z ratio from the first derivation. It therefore inherits that ratio's direction-dependence: $\overline{\Delta z}/\Delta z=e^{-2i\theta}$ along $\Delta z=r\,e^{i\theta}$ . Why: the increment does not cancel — it survives as a conjugate — so the quotient remembers the direction of approach.

Step 3. Test two paths for $\bar z$ . Along $\Delta z=t$ (real) the quotient is $+1$ ; along $\Delta z=it$ (imaginary) it is $-1$ . Different values, so $\frac{\mathrm{d}}{\mathrm{d}z}\bar z$ exists nowhere, despite $\bar z$ being continuous at every point of $\mathbb{C}$ . Why: this is the flat contradiction that proves continuity does not imply differentiability — the whole reason complex differentiability is a special, restrictive property.

Step 4. Interpret in components. Demanding the quotient agree along the $x$ - and $y$ -axes for a general $f=u+iv$ forces $u_x=v_y$ and $u_y=-v_x$ — the Cauchy–Riemann equations. A function passing this test throughout an open set is called analytic, and analyticity is where the power of the theory is stored. Why: the abstract "path-independent quotient" becomes a concrete pair of PDEs, the handoff to the next article.

Limiting cases

Every limit question is settled by comparing approaches; the table below is a menu of the outcomes you will meet, each with the reasoning that decides it.

Case	Result	Meaning
Limit exists after cancellation	$\lim_{z\to0}\dfrac{z^2}{z}=\lim_{z\to0}z=0$	The apparent $0/0$ removes: off the origin the function is $z$ , continuous everywhere, so every path gives $0$ .
Limit fails, direction-dependent	$\lim_{z\to0}\dfrac{\bar z}{z}=e^{-2i\theta}$ on $z=re^{i\theta}$	Value depends on the angle of approach; no single limit despite $\|\bar z/z\|=1$ being constant.
Removable point	$\lim_{z\to 2i}\dfrac{z^2+4}{z-2i}=4i$	Factor $(z-2i)(z+2i)$ cancels the zero denominator; the hole can be filled by defining the value $4i$ .
Pole (limit is $\infty$ )	$\lim_{z\to 0}\dfrac{1}{z}=\infty$	The modulus diverges along every path, a genuine limit on the Riemann sphere; the singularity is a pole.
Limit at infinity	$\lim_{z\to\infty}f(z)=L \iff \lim_{w\to0}f(1/w)=L$	Substitute $z=1/w$ ; approaching $\infty$ is approaching the north pole of the Riemann sphere.
Continuous, not differentiable	$f(z)=\|z\|^2=x^2+y^2$	A polynomial in $x,y$ , continuous on all of $\mathbb{C}$ , yet its difference quotient is path-dependent except at $z=0$ .

Common trapAgreement along the axes is not a limit

The commonest error is to treat a complex limit like a real two-sided limit: you check that $f(z)\to L$ along the real axis and along the imaginary axis, see the same $L$ , and declare victory. That is necessary but nowhere near sufficient. In the plane there are infinitely many paths — every ray $z=z_0+t\,e^{i\theta}$ , every parabola, every spiral — and the limit exists only if all agree. The map $f(z)=\dfrac{x y}{x^2+y^2}$ is the classic trap: $0$ along both axes yet $\tfrac12$ along $y=x$ , so no limit at the origin. Two coincident answers prove nothing; test a whole family. And beware the mirror error: $|f|$ continuous does not make $f$ continuous ( $\bar z/z$ again), and $f$ continuous never by itself makes $f$ differentiable.

Worked examples

1 · A removable singularity, fully evaluated

Evaluate $\displaystyle\lim_{z\to 2i}\frac{z^2+4}{z-2i}$ , or show it fails.

Step 1. Test the point directly. Substituting $z=2i$ gives $\dfrac{(2i)^2+4}{2i-2i}=\dfrac{-4+4}{0}=\dfrac{0}{0}$ — indeterminate, so the naive substitution is useless and we must look at the structure. A $0/0$ form is a signal that numerator and denominator share a zero, which may cancel.

Step 2. Factor the numerator over $\mathbb{C}$ . Since $z^2+4=(z-2i)(z+2i)$ , the quotient is $\dfrac{(z-2i)(z+2i)}{z-2i}=z+2i$ for every $z\neq 2i$ ; the offending factor cancels exactly, leaving a polynomial.

Step 3. Take the limit of the survivor. The function $z+2i$ is continuous everywhere, so the limit is just its value at $z=2i$ : $2i+2i=\mathbf{4i}$ , and no path-dependence can arise.

Step 4. Interpret. The original function was undefined at $z=2i$ but has a perfectly good limit there; the singularity is removable, and defining $f(2i)=4i$ makes $f$ continuous. The limit is $4i$ , and $z=2i$ is a removable singularity of $(z^2+4)/(z-2i)$ .

2 · A genuine ε–δ construction with numbers

For $f(z)=z^2$ at $z_0=1-i$ , find an explicit $\delta$ that guarantees $|f(z)-f(z_0)|<\varepsilon$ for the tolerance $\varepsilon=0.01$ .

Step 1. Locate the target value and the constant. Here $f(z_0)=(1-i)^2=1-2i+i^2=-2i$ , and $|z_0|=\sqrt{1^2+(-1)^2}=\sqrt2\approx 1.414$ . The controlling constant from the continuity derivation — the worst-case size of the nuisance factor $|z+z_0|$ — is $1+2|z_0|=1+2\sqrt2\approx 3.828$ .

Step 2. Apply the formula $\delta=\min\!\big(1,\ \varepsilon/(1+2|z_0|)\big)$ . The second entry is $\varepsilon/(1+2\sqrt2)=0.01/3.828\approx 0.00261$ ; since $0.00261<1$ , the minimum is $\boldsymbol{\delta\approx 0.00261}$ .

Step 3. Verify the guarantee. For any $z$ with $|z-(1-i)|<0.00261$ we have $|z^2-(-2i)|=|z-z_0|\,|z+z_0|<0.00261\times 3.828\approx 0.0100$ . The bound is met with essentially no slack, confirming the choice is sharp. A radius of $\delta\approx 0.00261$ around $z_0=1-i$ keeps $f(z)=z^2$ within $\varepsilon=0.01$ of $-2i$ .

3 · A limit that does not exist, from a two-variable trap

Does $\displaystyle\lim_{z\to 0}\frac{(\operatorname{Re}z)(\operatorname{Im}z)}{|z|^2}$ exist? Here the function is $f(z)=\dfrac{xy}{x^2+y^2}$ .

Step 1. Approach along the axes. Along the real axis $y=0$ , so $f=\dfrac{x\cdot 0}{x^2}=0$ ; along the imaginary axis $x=0$ , so $f=\dfrac{0\cdot y}{y^2}=0$ . Both axes give $0$ — and testing only these is the exact trap the warning box names.

Step 2. Approach along the diagonal $y=x$ . Now $f=\dfrac{x\cdot x}{x^2+x^2}=\dfrac{x^2}{2x^2}=\dfrac12$ , independent of how close we are to $0$ . A third path gives $\tfrac12\neq 0$ .

Step 3. Generalise and conclude. Along $y=mx$ the function is $\dfrac{m x^2}{(1+m^2)x^2}=\dfrac{m}{1+m^2}$ , a different value for every slope $m$ . Since distinct paths give distinct answers, the limit does not exist. No limit: $f=xy/(x^2+y^2)$ returns $m/(1+m^2)$ along $y=mx$ , sweeping $[-\tfrac12,\tfrac12]$ as the direction turns.

Physics of the portrait

The theory of complex functions was founded by Cauchy and given its geometry by Riemann, but the airtight language in which we now state a limit was forged by one man drilling it into a generation of students in Berlin.

Origins · 1 minKarl Weierstrass and how this was discovered

Karl Weierstrass

1815–1897 · the ε–δ definition of limit and continuity, and the divorce of continuity from differentiability

Karl Weierstrass came to mathematics by an unlikely road: he spent his twenties as a provincial Prussian schoolteacher, publishing almost nothing, working on Abelian functions by candlelight after teaching gymnastics and penmanship by day. A single 1854 paper on Abelian integrals detonated his obscurity and won him, in 1856, a chair in Berlin, where for three decades his lectures set the standard of rigour for all of analysis. His obsession was to expel vague talk of "quantities that become infinitely small" from the calculus. In its place he put the static, quantifier-based $\varepsilon$ – $\delta$ definition of limit and continuity we still use: for every $\varepsilon$ there is a $\delta$ — no motion, no infinitesimals, only inequalities between fixed numbers. This is the machinery of equation (2), and it is Weierstrass's more than anyone's. He wielded it to shocking effect in 1872, exhibiting a function continuous at every point of the line yet differentiable at none — a curve that is all corners, no tangents anywhere — which demolished the belief that continuity and smoothness travel together and forced everyone to take the $\varepsilon$ – $\delta$ definitions literally. In complex analysis he built the whole edifice from convergent power series (the "Weierstrass approach", complementary to Cauchy's integrals and Riemann's geometry), proved the theorem on the wild behaviour near an essential singularity that bears his and Casorati's names, and trained a remarkable school — Kovalevskaya, Schwarz, Frobenius, Mittag-Leffler among them. His monument is the guarantee that when we write $\lim_{z\to z_0}f(z)=L$ , we mean something exact.

History

1821

— Cauchy's Cours d'analyse grounds limits and continuity in inequalities rather than infinitesimals

1851

— Riemann's dissertation gives complex functions their geometric and conformal-mapping viewpoint

1861

— Weierstrass states the static ε–δ definition of limit and continuity in his Berlin lectures

1872

— Weierstrass exhibits a continuous nowhere-differentiable function, severing continuity from differentiability

1876

— the Casorati–Weierstrass theorem reads essential singularities off the limit behaviour of a function

Going further

Texts. Stone & Goldbart, Mathematics for Physics (CUP, 2009), Ch. 17–18 · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 11 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 24 · Ahlfors, Complex Analysis, 3rd ed., Ch. 1–2 · Churchill & Brown, Complex Variables and Applications, Ch. 2 · Needham, Visual Complex Analysis, Ch. 1–4 (for the geometric picture).

Provenance. All $\varepsilon$ – $\delta$ definitions and the component-limit equivalence (3) are standard (Ahlfors Ch. 2, Churchill–Brown Ch. 2). A verifier should recompute: (1) the radial values $e^{-2i\theta}$ for $\bar z/z$ , giving $+1,-i,-1,+i$ at $\theta=0,45^\circ,90^\circ,135^\circ$ ; (2) $\delta=\min(1,\varepsilon/(1+2|z_0|))$ for $z^2$ , and the numeric case $z_0=1-i,\ \varepsilon=0.01\Rightarrow\delta\approx0.00261$ ; (3) $z^2+4=(z-2i)(z+2i)$ giving the removable limit $4i$ ; (4) the trap $xy/(x^2+y^2)\to m/(1+m^2)$ along $y=mx$ . Every result is exact algebra with no empirical constants. Cross-check $2z_0+\Delta z$ (for $z^2$ ) and $\overline{\Delta z}/\Delta z$ (for $\bar z$ ) against Stone & Goldbart Ch. 17.

Prerequisites. Complex Numbers and the Complex Plane. Leads to. Analytic Functions and the Cauchy–Riemann Equations · Elementary Complex Functions · Complex Integration and Contour Integrals.

Ask the tutor: ask me to walk you through the path-dependence test on any specific function, or to build the explicit $\delta$ for a continuity proof step by step.

Is a complex limit really just two real limits stacked together?

Essentially yes, and that equivalence — equation (3) — is your most reliable tool. Because $\mathbb{C}$ with the modulus distance is isometric to $\mathbb{R}^2$ , saying $f(z)\to a+ib$ is exactly saying $u(x,y)\to a$ and $v(x,y)\to b$ as two-variable real limits, so every $\mathbb{R}^2$ technique — testing paths, squeeze bounds, polar coordinates — applies verbatim. The one thing the split hides is the multiplicative structure of $\mathbb{C}$ : the difference quotient divides by a complex number, an operation with no counterpart in a bare $\mathbb{R}^2$ , and it is there that genuinely complex phenomena (analyticity, the Cauchy–Riemann equations) enter. For limits and continuity alone, "two real limits" is the whole story.

Why does the value f(z₀) not appear in the definition of the limit?

Look at the clause $0<|z-z_0|$ in equation (2): the strict left inequality deliberately deletes the centre, so the limit is computed from the behaviour of $f$ near $z_0$ but never at it. A limit describes the trend as you close in, and that trend must be knowable even when $f(z_0)$ is undefined or singular — as in the removable case $(z^2+4)/(z-2i)$ , which has no value at $2i$ yet a clean limit of $4i$ . Continuity is the separate, stronger condition (equation 4) that additionally demands $f(z_0)$ exist and equal the limit. Keeping the two apart is exactly what lets you classify singularities: same limit machinery, different answers about the point itself.

How can |z̄/z| have a limit while z̄/z does not?

Because modulus throws away the phase, and the phase is exactly what misbehaves. Along $z=r e^{i\theta}$ the quotient is $\bar z/z=e^{-2i\theta}$ : its modulus is $|e^{-2i\theta}|=1$ for every direction, a perfectly constant — hence convergent — quantity, but its argument $-2\theta$ takes a different value for every direction, so the complex number itself never settles. This is a special case of a general fact: continuity of $|f|$ is strictly weaker than continuity of $f$ , because $|f|$ is blind to where $f$ points. Whenever you see a "limit" argued purely from moduli, be suspicious — you must also check that the phase converges, or the whole limit can silently fail while the modulus looks healthy.

What is the difference between continuity and complex differentiability, really?

Continuity asks that $f(z)\to f(z_0)$ — the value varies without jumps. Differentiability asks the far harder question that the rate of change $\big(f(z)-f(z_0)\big)/(z-z_0)$ approach a single number regardless of direction. Division by a complex number rotates as well as scales, so demanding one answer for all directions is a genuine constraint. The map $\bar z$ shows the gap starkly: continuous everywhere (it just reflects across the real axis), yet its quotient $\overline{\Delta z}/\Delta z$ gives $+1$ along the real direction and $-1$ along the imaginary one, so it is differentiable nowhere. That gap, written in components, is the Cauchy–Riemann equations.

How do you make sense of a limit as z → ∞?

Add a single "point at infinity" to the plane, turning it into the Riemann sphere: the finite plane wraps onto a sphere with $z=0$ at the south pole and $\infty$ at the north. Approaching $\infty$ then just means approaching the north pole, and the clean way to compute it is the substitution $w=1/z$ , so $\lim_{z\to\infty}f(z)=L$ is defined to mean $\lim_{w\to0}f(1/w)=L$ with the ordinary $\varepsilon$ – $\delta$ definition in $w$ . This is not a hand-wave: on the sphere $\infty$ is an honest point, and a rational function like $(3z^2+1)/(z^2-5)$ has the perfectly good limit $3$ there. The sphere also lets a pole be a genuine limit — $1/z\to\infty$ as $z\to0$ — rather than a mere failure to converge.

Which everyday functions are continuous, and where do they break?

Polynomials are continuous on all of $\mathbb{C}$ ; rational functions everywhere except the zeros of their denominator; and $e^z$ , together with $\cos z,\sin z,\cosh z,\sinh z$ built from it, on the whole plane. Sums, products, quotients (nonzero denominator), and compositions of continuous functions are again continuous, so almost anything in closed form is continuous on its natural domain. The interesting breakage lives at branch points and branch cuts: $\log z$ and $\sqrt z$ cannot be made continuous on a loop encircling the origin, because the value shifts by $2\pi i$ (or a sign) each time around, so one must cut the plane to keep them single-valued. Those cuts are where a limit becomes genuinely two-valued, and tracking them is a large part of the working craft of complex analysis.