# Thread: Finding a function that works

1. ## Finding a function that works

Hey guys, so i have the following:

$\displaystyle f''(a) = \lim_{h \to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$

and I'm trying to figure out a function that works for that limit, but doesn't have a second derivative at $\displaystyle a$... been scratching my head over this one for a while lol thanks!

2. edit2: wait, no, I was right lol. Let a=0 and try $\displaystyle x^2\sin (\frac1x)$.

3. $\displaystyle f(a)=0$

$\displaystyle f(x)=1$ if $\displaystyle x<a$

$\displaystyle f(x)=-1$ if $\displaystyle x>a$

The limit clearly exists and is zero, for the numerator is identically zero. But $\displaystyle f(x)$ is discontinuous at $\displaystyle a$ and hence has no derivatives there.