Given some function $f: I \subseteq\mathbb R \rightarrow \mathbb R$, Which is differentiable twice at some point $a\in I$.
Can one say that there is a region around the point where the function is differentiable twice, without any other information?
so I assume that its true because if we look at the first derivative which is:
$lim_{h\rightarrow0} \frac {f(a+h)-f(a)}h $ then obliviously we can "take" h to be smaller as we want and then I can assume that if the limit exists then it exists at some region of that point a.
so I guess the same goes for the second derivative.
Best Answer
Let $f\colon\mathbb R\longrightarrow\mathbb R$ a continuous function which is differentable nowhere, let $F$ be a primitive of $f$ and let $g(x)=x^2F(x)$. Then $g$ is differentiable: $g'(x)=2xF(x)+x^2f(x)$. And $g'$ is differentiable at $0$. But that's the only point of $\mathbb R$ at which $g'$ is differentiable.