Continuity — Definition, Formula & Examples

Continuity means a function has no breaks, holes, or jumps at a given point — you can draw through it without lifting your pencil. A function is continuous at a point when the limit exists there and equals the actual function value.

A function $f$ is continuous at $x = c$ if and only if three conditions hold: (1) $f(c)$ is defined, (2) $\lim_{x \to c} f(x)$ exists, and (3) $\lim_{x \to c} f(x) = f(c)$ . A function is continuous on an interval if it is continuous at every point in that interval.

Key Formula

\lim_{x \to c} f(x) = f(c)

Where:

$f$ = The function being tested for continuity
$c$ = The specific x-value where continuity is being checked

How It Works

To check continuity at a specific point

x = c

, verify the three conditions in order. First, confirm the function is defined at

c

. Second, compute the left-hand and right-hand limits; if they are equal, the two-sided limit exists. Third, check whether that limit equals

f(c)

. If any condition fails, the function is discontinuous at that point.

Worked Example

Problem: Determine whether f(x) = (x² − 4)/(x − 2) is continuous at x = 2.

Condition 1: Check if f(2) is defined. Substituting gives 0/0, which is undefined.

f(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0} \text{ (undefined)}

Condition check: Even though the limit exists (factor and simplify to get the limit equals 4), the first condition already fails.

\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \implies \lim_{x \to 2} f(x) = 4

Answer: f is not continuous at x = 2 because f(2) is undefined. There is a removable discontinuity (hole) at that point.

Why It Matters

Continuity is a prerequisite for major calculus theorems. The Intermediate Value Theorem guarantees a continuous function hits every value between its endpoints, which is used to prove roots exist. The Extreme Value Theorem guarantees a continuous function on a closed interval has a maximum and minimum — essential for optimization problems in AP Calculus and engineering.

Common Mistakes

Mistake: Assuming a function is continuous just because the limit exists at a point.

Correction: The limit existing is only one of three conditions. You must also verify that f(c) is defined and that the limit equals f(c). A hole in the graph means the limit exists but the function value is missing or different.