Finding a function that works

**Kipster1203** · April 25th 2010, 05:07 PM

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!

**maddas** · April 25th 2010, 05:38 PM

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

**becko** · April 25th 2010, 07:07 PM

$f(a)=0$

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

$f(x)=-1$ 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.

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