Increasing function

A function y=f(x) is said to be increasing on an interval I=(a,b) if, for any two points x₁, x₂ in the interval with x₁<x₂, we have $$ f(x_1) \le f(x_2) $$. It is called strictly increasing if instead $$ f(x_1) < f(x_2) $$ holds.

A strictly increasing function is sometimes described as being “increasing in the strict sense.”

Both increasing and strictly increasing functions belong to the broader class of monotone functions.

Note. A function is monotone on an interval of its domain if, throughout that interval, it is either always increasing or always decreasing.

Example

Consider the function

$$ y=x^2 $$

This function is strictly increasing on the interval (1,5), since for any two points x₁<x₂ in that interval we have f(x₁)<f(x₂).

On the other hand, the same function is not increasing on the interval (-5,-1).

Note. The monotonicity of a continuous, differentiable function can also be analyzed using the first derivative test.

And so forth.