Got this from my Real Analysis problem set:
Suppose $f(x)$ continuous on some open interval $I$, and $c$ is maximum point for $f(x)$ inside this interval. Is it true that that $f(x)$ is increasing immediately in the left of $c$ and immediately decreasing in the right of $c$?
(The constant function is not a counterexample because it's considered to both increasing and decreasing.)
I really think so. Am I wrong?
If the function strictly incraeses after $c$, so $c$ cannot be a maximum.
Am I not seeing something important. This seems obvious :/
Any help would be awesome, thanks in advance!