Increasing Functions

A function y = f(x) is called increasing on an interval I = (a, b) if, for any two points x₁ and x₂ in the interval such that x₁ < x₂, the following holds: $$ f(x_1) \le f(x_2) $$. It is called strictly increasing if: $$ f(x_1) < f(x_2) $$

A strictly increasing function is sometimes referred to as increasing in the strict sense.

Both increasing and strictly increasing functions are part of the broader family of monotonic functions.

Note: A function is said to be monotonic over an interval of its domain if it is either consistently increasing or consistently decreasing throughout that interval.

Example

Consider the function:

$$ y = x^2 $$

This function is strictly increasing on the interval (1, 5), since for any two points x₁ and x₂ with x₁ < x₂, the inequality f(x₁) < f(x₂) holds true.

However, the same function is not increasing on the interval (-5, -1).

Note: If a function is continuous and differentiable, its increasing or decreasing behavior can also be analyzed using the first derivative.

And so on.