continuity

A function has no breaks, jumps, or holes at a point — its limit there equals its actual value.

$f$ is continuous at $x=a$ when three things all hold: $f(a)$ is defined, $\lim_{x\to a} f(x)$ exists, and $\lim_{x\to a} f(x) = f(a)$ . Fail any one of the three and $f$ is discontinuous at $a$ .

Examples

Every polynomial is continuous everywhere; $f(x)=1/x$ is discontinuous only at $x=0$ .