Let $f$ be continuous at $b$ in a domain $[c,d]$.
Then does $[c,d]$ contain an interval $[a,b]$ such that $f$ is either increasing or decreasing on $[a,b]$?
For example, $y=x$ is continuous at $1$ in $[0,2]$ and $[0,2]$ contains $[0,1]$ on which $y=x$ is increasing.
I was wondering this is true in general.
