A function f is said to be monotonic increasing in (a,b) if in this interval, .

The function is said to be strictly monotonic increasing in (a,b) if in this interval, .

A function f is said to be monotonic decreasing in (a,b) if in this interval, .

The function is said to be strictly monotonic decreasing in (a,b) if in this interval, .

ExampleLet f be a strictly monotonic increasing function. Prove that f is an injection.

Solution

Let f be a strictly monotonic increasing function.

Let be such that .

Now which is a contradiction, and which also is a contradiction.

So and thus f is an injection.