Let $f$ be a real continuous function on a bounded interval $[a,b]$ such that every point $x$ in $[a,b)$ has a positive number $\delta$ where every point $t$ in $(x,x+\delta)$ satisfies $f(t)>f(x)$. Is $f$ increasing on $[a,b]$? (This question comes from trying to understand Rudin's proof of Theorem 7.21 in Real and Complex Analysis).
It's enough to show that $f$ is locally increasing, but this seems tricky. I feel that there might be transfinite induction involved, but I don't know much about this topic.
Suppose that it is not increasing. Then $f(x)>f(y)$ for some $x,y\in[a,b]$ such that $x<y$. Let $z\in[x,y]$ be such that $f(z)=\max f([x,y])$. Then $z<y$. Now apply your hypothesis to the point $z$.