Differentiability vs Continuity
Two of the most important properties a function can have at a point are continuity and differentiability. Continuity means that the graph of the function has no breaks, holes, or jumps at that point — you could draw it without lifting your pencil. Differentiability is a stronger condition: it means the function has a well-defined tangent line (a unique, finite slope) at that point. Understanding how these two ideas relate is fundamental to calculus. The central relationship is asymmetric and worth memorizing precisely: if a function is differentiable at a point, then it must be continuous there. However, the converse is false — a function can be continuous at a point without being differentiable there. Classic examples like the absolute value function |x| at x = 0 illustrate this perfectly: the graph is unbroken but has a sharp corner where no single tangent line exists. Mastering this distinction helps you predict where derivatives can and cannot exist, identify where optimization and tangent-line techniques apply, and understand why certain physical models break down at sharp transitions. It also deepens your appreciation for the logical structure of calculus, where stronger conditions imply weaker ones but not vice versa.
- Recall the Definition of Continuity
- Recall the Definition of Differentiability
- The Theorem: Differentiability Implies Continuity
- Why the Converse Fails: Sharp Corners
- Other Ways Differentiability Can Fail
- Summary of the Logical Relationship
Recall the Definition of Continuity
A function f is continuous at x = a if three conditions hold: f(a) is defined, the limit lim_{x→a} f(x) exists, and lim_{x→a} f(x) = f(a). Geometrically, the graph passes through (a, f(a)) with no gap or jump. Continuity is a necessary but not sufficient condition for differentiability.
Recall the Definition of Differentiability
A function f is differentiable at x = a if the following limit exists and is finite: f'(a) = lim_{h→0} [f(a+h) − f(a)] / h. This limit defines the slope of the tangent line. If the limit does not exist — because the left-hand and right-hand limits differ, or because the limit is infinite — the function is not differentiable at that point.
The Theorem: Differentiability Implies Continuity
Proof sketch: Suppose f is differentiable at a, so f'(a) exists. We want to show lim_{x→a} f(x) = f(a). Write f(x) − f(a) = [(f(x)−f(a))/(x−a)] · (x−a). As x→a, the first factor approaches f'(a) (a finite number) and the second factor (x−a) approaches 0. Therefore lim_{x→a} [f(x)−f(a)] = f'(a)·0 = 0, which means lim_{x→a} f(x) = f(a). So f is continuous at a. ∎
Why the Converse Fails: Sharp Corners
Consider f(x) = |x| at a = 0. The function is continuous: f(0) = 0 and lim_{x→0} |x| = 0. But the derivative limit is lim_{h→0} |h|/h. From the right, |h|/h = 1; from the left, |h|/h = −1. Since the one-sided limits differ, the derivative does not exist at x = 0. Continuity does not guarantee differentiability.
Other Ways Differentiability Can Fail
Beyond sharp corners, differentiability also fails at: (1) Cusps — where |f'(x)| → ∞ from both sides, e.g. f(x) = x^{2/3} at x = 0; (2) Vertical tangents — where the slope becomes infinite, e.g. f(x) = x^{1/3} at x = 0; (3) Discontinuities — any jump, hole, or infinite discontinuity automatically prevents differentiability because continuity is a prerequisite.
Summary of the Logical Relationship
In formal logic notation: Differentiable at a ⟹ Continuous at a. The contrapositive is equally useful: Not continuous at a ⟹ Not differentiable at a. If you discover a function has a jump discontinuity, you can immediately conclude no derivative exists there, without any further calculation.
The Key Equation
- f'(a) — Derivative of f at a (units of f per unit of x): The instantaneous rate of change of f at the point x = a; equals the slope of the tangent line if it exists.
- h — Increment (same as x): A small change in the input variable that approaches zero in the limit process.
- f(a+h) - f(a) — Change in output (same as f): The numerator of the difference quotient; measures how much the function value changes when x moves from a to a+h.
Worked Example
Problem. Let f(x) = |x − 2|. (a) Is f continuous at x = 2? (b) Is f differentiable at x = 2? Justify both answers using limits.
Answer. Continuous at x = 2; not differentiable at x = 2 because the left and right derivative limits are −1 and 1 respectively.
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to give rigorous epsilon-delta definitions of limits, continuity, and derivatives in his landmark textbook Cours d'analyse (1821). He formalized precisely the conditions under which a function is continuous and laid the groundwork for the proof that differentiability implies continuity — converting these ideas from intuitive geometric notions into strict logical statements. Cauchy published an astonishing 789 mathematical papers and books during his lifetime — averaging roughly one new publication every six days — making him one of the most prolific mathematicians in history.
Related Concepts
- The Formal Definition of a Limit — Differentiability is itself defined as a limit of the difference quotient, so a solid understanding of limits is an essential prerequisite for this topic.
- Types of Discontinuity — Since discontinuity guarantees non-differentiability, recognizing jump, removable, and infinite discontinuities is directly useful for identifying where derivatives cannot exist.
- Basic Differentiation Rules — Once differentiability conditions are understood, these rules (power rule, product rule, etc.) provide efficient methods to compute derivatives where they do exist.
- The Mean Value Theorem — The Mean Value Theorem requires both continuity on a closed interval and differentiability on the open interior — a direct application of the hierarchy between these two properties.