Theorem 3 A continuous function defined on a closed interval is one-to-one if and only if it is strictly monotone. Suppose f : E → R is a strictly monotone function defined on a set E ⊂ R. Then f is one-to-one on E so that the inverse function f -1 is a well defined function on f (E).

Are continuous functions monotonic?

Monotone functions are differentiable almost everywhere. So if a function is monotonic on an interval it is differentiable on a set of positive measure. However there are continuous nowhere differentiable functions so it isn't true that continuous functions are monotonic in a (one sided) neighborhood of every point.

How do you know if a function is monotonic?

A function that is completely increasing or completely decreasing on the given interval is called monotonic on the given interval. Usually, by looking at the graph of the function one can say whether the function is increasing or decreasing or neither.