## MA101.6 Monotonic Functions

A function f is said to be monotonic increasing in (a,b) if $\forall x_1,x_2$ in this interval, $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$.

The function is said to be strictly monotonic increasing in (a,b) if $\forall x_1,x_2$ in this interval, $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$.

A function f is said to be monotonic decreasing in (a,b) if $\forall x_1,x_2$ in this interval, $x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)$.

The function is said to be strictly monotonic decreasing in (a,b) if $\forall x_1,x_2$ in this interval, $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$.

Example

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

Solution

Let f be a strictly monotonic increasing function.
Let $x_1,x_2 \in D(f)$ be such that $f(x_1) = f(x_2)$.
Now $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$ which is a contradiction, and $x_2 < x_1 \Rightarrow f(x_2) < f(x_1)$ which also is a contradiction.
So $x_1 = x_2$ and thus f is an injection.