# Thread: Finding a function that works

1. ## Finding a function that works

Hey guys, so i have the following:

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

and I'm trying to figure out a function that works for that limit, but doesn't have a second derivative at $a$... 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 $x^2\sin (\frac1x)$.

3. $f(a)=0$

$f(x)=1$ if $x

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

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