Let $C([a,b])$ be the space of continuous functions on $[a,b]$ in the uniform topology.Suppose that $[c,d]\subset[a,b].$ Show that
$M=${$f\in C([a,b]):f$ is monotone on $[c,d]$} is a closed set.
I don't have any specific idea, i tried some contradiction but doesn't work. Next I tried showing compliment is open, does not work.