To prove the existence of nowhere monotone continuous functions, one step is to show that for any interval $I \subset [0,1]$ of positive length, the set $A_I= \{f \in C([0,1]): f|_I$ is increasing$\}$ is closed (i.e. see http://www.apronus.com/math/nomonotonic.htm). I am not sure how to do prove this.
My idea is that if $(f_n) \in A_I$ converges to $f$, then for all $x
Thank you.