A function increases on an Interval if for all , where . Conversely, a function decreases on an Interval if for all with .
If the Derivative of a Continuous Function satisfies on an Open Interval , then is increasing on . However, a function may increase on an interval without having a derivative defined at all points. For example, the function is increasing everywhere, including the origin , despite the fact that the Derivative is not defined at that point.
See also Decreasing Function, Derivative, Nondecreasing Function, Nonincreasing Function