Functions strictly monotone increasing and monotone increasing

Question

How can I gather if a function is strictly monotone increasing thus is monotone increasing? Only using the two definitions

Answer 1

Let $A $ be a subset of $\Bbb R $.

If $f $ is strictly increasing at $A $, then $$(\forall (x,y)\in A^2)\;(x\ne y\implies \frac {f (x)-f (y)}{x-y}>0)$$

and $$(\forall (x,y)\in A^2)\;\;(x\ne y \implies \frac {f (x)-f (y)}{x-y}\ge 0)$$

$\implies \; f $ is increasing at $A $.

Answer 2

If a function $f$ is strictly monotone increasing then for $x, y$ such that $x < y$, $f(x) < f(y)$, so $f(x) \le f(y)$ so $f$ is monotone increasing.

Answer 3

For $\Delta x = x' - x \gt 0$, and
$\Delta f(x) = f(x') - f(x)$

$\Delta f(x) \gt 0$ implies strictly increasing

$\Delta f(x) \lt 0$ implies strictly decreasing

$\Delta f(x) \geq 0$ implies increasing

$\Delta f(x) \leq 0$ implies decreasing

In other words, if $f'(x)$ does not change sign over the interval, then the function is monotone.