A function is continuous at a point when its nearby values approach the value the function actually has there. Formally, $f$ is continuous at $a$ when

$$\lim_{x\to a}f(x)=f(a).$$

This requires three things: $f(a)$ must exist, the limit must exist, and the two values must agree.

Common discontinuities

A removable discontinuity occurs when the limit exists but the function is missing or has the wrong value at the point. For example,

$$f(x)=\frac{x^2-1}{x-1}$$

has a removable discontinuity at $x=1$ if it is left undefined there.

A jump discontinuity occurs when the left- and right-hand limits are different. An infinite discontinuity occurs when the function grows without bound near the point, as $1/x$ does near $0$.

Continuity on intervals

A function is continuous on an interval when it is continuous at every interior point, with the appropriate one-sided condition at included endpoints.

Polynomials are continuous everywhere, while rational functions are continuous wherever their denominators are nonzero. Exponential, logarithmic and trigonometric functions are continuous on their natural domains.

Why continuity matters

Continuity formalizes the idea that small changes in input do not produce sudden breaks in output. It also allows limits to pass through many familiar functions and is a key hypothesis in theorems throughout calculus.