# Second differentiability

• Feb 17th 2010, 12:16 AM
jackie
Second differentiability
I'm kinda stuck on this problem. Would someone give me a hand?
Suppose $\displaystyle f:[a,b] \rightarrow R$ is differentiable and $\displaystyle f'$ is differentiable at $\displaystyle t \in (a,b)$. Show that
$\displaystyle f''(t)=\lim_{h\to 0} \frac{f(t+h)-2f(t)+f(t-h)}{h^2}$
Is there a function that satisfies this limit but $\displaystyle f'$ is not differentiable at $\displaystyle t$?
• Feb 17th 2010, 12:56 AM
chisigma
For the function...

$\displaystyle f(t) = t^{2}\cdot sgn(t)$ (1)

... in $\displaystyle t=0$ the limit exists and it is equal to 0 but its derivative is not differenziable in that point...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Feb 17th 2010, 09:55 PM
Drexel28
Quote:

Originally Posted by jackie
I'm kinda stuck on this problem. Would someone give me a hand?
Suppose $\displaystyle f:[a,b] \rightarrow R$ is differentiable and $\displaystyle f'$ is differentiable at $\displaystyle t \in (a,b)$. Show that
$\displaystyle f''(t)=\lim_{h\to 0} \frac{f(t+h)-2f(t)+f(t-h)}{h^2}$
Is there a function that satisfies this limit but $\displaystyle f'$ is not differentiable at $\displaystyle t$?

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