Increasing Function

A function $f(x)$ increases on an Interval $I$ if $f(b)>f(a)$ for all $b>a$, where $a,b\in I$. Conversely, a function $f(x)$ decreases on an Interval $I$ if $f(b)<f(a)$ for all $b>a$ with $a,b\in I$.

If the Derivative $f'(x)$ of a Continuous Function $f(x)$ satisfies $f'(x)>0$ on an Open Interval $(a,b)$, then $f(x)$ is increasing on $(a,b)$. However, a function may increase on an interval without having a derivative defined at all points. For example, the function $x^{1/3}$ is increasing everywhere, including the origin $x=0$, despite the fact that the Derivative is not defined at that point.

See also Decreasing Function, Derivative, Nondecreasing Function, Nonincreasing Function