Continuity at a Point
Continuity is one of the most fundamental ideas in calculus. Intuitively, a function is continuous at a point if you can draw its graph through that point without lifting your pencil. More precisely, continuity at a point requires that the function is defined there, that the limit exists there, and that the two values agree. When any of these conditions fail, the function has a discontinuity at that point. Understanding continuity at a point is essential before studying derivatives, because differentiability actually implies continuity — every differentiable function is continuous, though the converse is not always true. Recognising where functions break down, and why, builds the analytical habits needed throughout calculus and beyond, from physics to economics. In this article you will learn the formal three-part definition of continuity at a point, how to verify continuity step by step, how to classify common types of discontinuities (removable, jump, and infinite), and how to apply these ideas to piecewise and rational functions. Worked examples and diagrams will guide you through each idea.
- Condition 1 — The function must be defined at the point
- Condition 2 — The limit must exist at the point
- Condition 3 — The limit must equal the function value
- Checking continuity — a systematic approach
- Types of discontinuity
Condition 1 — The function must be defined at the point
For a function f to be continuous at x = a, the value f(a) must exist. If a is not in the domain of f, continuity at a is automatically impossible. For example, f(x) = 1/x is not defined at x = 0, so it cannot be continuous there.
Condition 2 — The limit must exist at the point
The two-sided limit lim_{x→a} f(x) must exist and be finite. This means the left-hand limit lim_{x→a⁻} f(x) and the right-hand limit lim_{x→a⁺} f(x) must both exist and be equal to each other. If they differ, the function has a jump discontinuity at x = a.
Condition 3 — The limit must equal the function value
Finally, the limit and the function value must agree: lim_{x→a} f(x) = f(a). A function can satisfy Conditions 1 and 2 but still fail Condition 3 — for instance, if f(a) has been redefined to a different value. This creates a removable discontinuity, sometimes called a 'hole' in the graph.
Checking continuity — a systematic approach
To verify continuity at x = a: (i) Find f(a) and confirm it exists. (ii) Compute the left- and right-hand limits separately if the function is piecewise, otherwise compute the two-sided limit using limit laws or direct substitution. (iii) Check whether lim_{x→a} f(x) = f(a). If all three hold, write 'f is continuous at x = a.' If any fail, identify and name the type of discontinuity.
Types of discontinuity
A removable discontinuity occurs when the limit exists but does not equal f(a) (or f(a) is undefined) — the graph has a single isolated hole. A jump discontinuity occurs when the left- and right-hand limits exist but are unequal — the graph jumps from one value to another. An infinite (essential) discontinuity occurs when at least one side of the limit grows without bound, as near a vertical asymptote. Rational functions, piecewise functions, and absolute-value functions are common sources of all three types.
The Key Equation
- f(a) — Function value at a (same units as f): The actual output of the function when the input is exactly a; must be defined.
- \lim_{x \to a} f(x) — Two-sided limit (same units as f): The value that f(x) approaches as x gets arbitrarily close to a from both sides; must exist and be finite.
- a — Point of interest (same units as x): The specific x-value at which continuity is being tested.
- \iff — If and only if (—): Logical biconditional: the condition on the left is equivalent to the condition on the right.
Worked Example
Problem. Let f be defined by f(x) = (x² − 4)/(x − 2) for x ≠ 2, and f(2) = 5. Is f continuous at x = 2? If not, identify the type of discontinuity.
Answer. f is NOT continuous at x = 2. It has a removable discontinuity there because the limit equals 4 but f(2) = 5.
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to give a rigorous formal definition of continuity and limits in his 1821 textbook Cours d'analyse. He replaced vague geometric intuitions with precise algebraic conditions, establishing the foundations of modern real analysis. His definition of continuity — that f is continuous at a point if an infinitely small change in x produces an infinitely small change in f(x) — evolved into the epsilon-delta definition that underpins all of calculus today. Cauchy published over 800 research papers and books during his lifetime — more than any other mathematician in history at the time — making him one of the most prolific scientists who ever lived.
Related Concepts
- The Limit of a Function — Continuity at a point is defined in terms of limits; you must understand how to compute and interpret limits before verifying continuity.
- One-Sided Limits — Jump discontinuities are detected by comparing left-hand and right-hand limits, so one-sided limits are essential tools for continuity analysis.
- Differentiability at a Point — Differentiability implies continuity, making continuity a necessary condition for a derivative to exist at a point.
- Continuity on an Interval — Once continuity at individual points is understood, the concept extends naturally to open and closed intervals, enabling theorems like the Intermediate Value Theorem.