Suppose we have a continuous $f: \mathbb{R} \to \mathbb{R}$, we know the definition of a monotonically increasing function is for $x,y \in \mathbb{R}$, if $x < y$ then $f(x) < f(y)$. I know that if $f(x) < f(y)$ then $x < y$ is not the contra-positive to the statement, so they must not be equivalent. But I am having trouble coming up with a counter example.
If we have f(x) = x, then indeed the two statements are true, what function could we use to show that they are not?
There is no counter example. Suppose you are given $f(x) < f(y)$. Now either $x=y$ or $x>y$ or $x<y$. But if $x=y$then $f(x)=f(y)$. If $x>y$ then monotonicity gives us $f(x)>f(y)$.