Increasing and Decreasing Functions

increasing decreasing functionsA 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.

function y=x^2

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)`.

non-increasing non-decreasing functionsThe 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.