real analysis - Showing that the set of functions in $C(I)$ which are monotone on some nontrivial subinterval of $I$ is of first category in $C(I)$.

Let $I = [0,1]$ and let $C(I)$ be the metric space of continuous functions on $I$ with the $L^{\infty}$ norm. I am trying to show that the set of functions in $C(I)$ which are monotone on some nontrivial subinterval of $I$ is of first category in $C(I)$.

I believe that this means that a ‘generic’ continuous function on $[0,1]$ is not monotone on any nontrivial subinterval.

I'd like to use the Baire Category Theorem, but frustratingly am running into troubles with it. Is the Baire Category Theorem the right approach? Is there another way to do this? Thanks!

Answer

What can you say about the subset $C_{r,s}^{+}(I)$ of $C(I)$ consisting of functions which are non-decreasing on $[r,s]\subseteq I$?