I've a question on the study of the monotony of a function.
There is this theorem:
Let $f(x)$ differentiable in a interval $I$ and $f'(x_0)=0$ (with $x_0 \in I$).
If $f'(x)\geq 0$ from the left of $x_0$ and $f'(x)\leq 0$ from the right of $x_0$, then $x_0$ is a maximum point for the function $f(x)$. (In the opposite case it is a minimum point).
When I'm looking for the points of maxima and minima for the function I must check also the points in which $f'(x)$ is not differentiable.
What I wonder is: in that case is it correct to say that
Let $f(x)$ differentiable in a interval $I-\big\{x_0\big\}$ (with $x_0 \in I$).
If $f'(x)\geq 0$ from the left of $x_0$ and $f'(x)\leq 0$ from the right of $x_0$, then $x_0$ is a maximum point for the function $f(x)$. (In the opposite case it is a minimum point).
?
Thanks a lot for your help
This is not the case. Take, for example, $f(x)=x$ for $x<0$, $f(x)=-x+1$ for $x>0$ and $f(0)=1/2$.
You will need, at least, continuity on $x_0$ (and continuity con $x_0$ plus differentiability on $I$ implies continuity on $I$).