If f(x) = Mod(x), 0<Mod(x)<=2
1 x=0
Is there any local extremum at x=0.
By "mod(x)", do you mean "absolute value of x", |x|?
If so, your function is f(x)= |x| for $\displaystyle -2\le x< 0$ and $\displaystyle 0< x\le 2$ while f(0)= 1. Yes, x= 0 is a local maximum since f(0)= 1 is larger than f(x) for x close to, but not equal to 0.
If you mean $\displaystyle f(x)=\begin{Bmatrix} |x| & \mbox{ if }& 0<|x|<2\\1 & \mbox{if}& x=0\end{matrix}$ , then there is a local maximum at $\displaystyle x=0$ because $\displaystyle f(0)\geq f(x)$ in a neighborhood of $\displaystyle 0$ .
Edited: Sorry, I didn't see HallsofIvy post.