# Increasing and Decreasing Functions

A function f is called increasing on interval I if f(x_1)<f(x_2) whenever x_1<x_2 in I.

A function f is called decreasing on interval I if f(x_1)>f(x_2) whenever x_1>x_2 in I.

A function f is called non-decreasing on interval I if f(x_1)<=f(x_2) whenever x_1<x_2 in I.

A function f is called non-icreasing on interval I if f(x_1)>f(x_2) whenever x_1>x_2 in I.

All above types of functions have common name. They are called monotonic.

It is important to understand that in definition of increasing f(x_1)<f(x_2) should hold for any x_1 and x_2 such that x_1<x_2. Same can be said about definition of decreasing, non-decreasing and non-increasing functions.

Consider function on the figure. It is increasing on interval [a,b] and [c,d], and decreasing on interval [b,c].

Function f(x)=x^2 is increasing on interval [0,oo) and decreasing on interval (-oo,0).

The only difference between increasing and non-decreasing, decreasing and non-increasing is that function can take constant value on some subinterval(s) of I. See figure to the right.