Let $f$ be any continuous monotonically increasing function with property that
$f(x) \in\mathbb Z\Rightarrow x \in\mathbb Z$
In the related proof, it says,
if $\lfloor x \rfloor < x$ then $f(\lfloor x \rfloor) < f(x)$
I believe the inequality should be $f(\lfloor x \rfloor) \le f(x)$ , since $f$ is not strictly increasing.
how we coclude that the function is strictly increasing?