Continuity

A function which is continuous is one in which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. Continuity is more rigorously defined as follows:

$f(x)$ is continuous at $x = c$ if and only if:

1. $f(c)$ exists, proving that a point exists in that function at $c$
2. $\lim_{x \to c} f(x)$ exists, proving that both sides of the function are approaching a specific value (if it is not possible to take a 2-sided limit, it may be necessary to do 2 1-sided limits and prove they are equal to each other first)
3. $f(c)=\lim_{x\to c}f(x)$ , proving the function is approaching the same value as the point is at

All three parts of the continuity test must be proven true in order for the function to be continuous.

Continuity may be considered important since if a function is not continuous, it is non-differentiable at that point as well. However, if the function is continuous, that does not necessarily mean it is differentiable there, although it may be.

All polynomial functions are continuous.