$ f'(x) \leq f'\left(x + \frac{1}{n} \right) $ for all $ x \in \mathbb{R} $, $ n \in \mathbb{N} $. Show that $ f' $ is continuous.

Romanian Math Olympiad 1998

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that$f'(x) \leq f'\left(x + \frac{1}{n} \right)$for all $x \in \mathbb{R}$ and all positive integers $n$. Prove that $f'$ is continuous.

The Solution Given

Consider the following function:$$f_n(x) = n \left( f\left(x + \frac{1}{n} \right) - f(x) \right).$$

By the hypothesis, $f_n$ has a nonnegative derivative; therefore it is increasing. Thus, if $x < y$, we have $f_n(x) \leq f_n(y)$ for all $n$. By making $n \rightarrow \infty$, we conclude that $f'(x) \leq f'(y)$. Thus, $f'$ is increasing. Using Darboux's theorem, we conclude that $f'$ is continuous. $\square$

My Doubt

why did we define $f_n(x)$? I mean what I thought was The given condition $f'(x) \leq f'\left(x + \frac{1}{n} \right)$ implies $f'$ is increasing and since $f$ is continous and differentiable by Darboux's Theorem it has IVP and any Increasing function with IVP is continous. I couldn't get the motivation of defining $f_n(x)$. Please clear my doubt.

Thanks for reading!