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I am looking for a function $f:[a,b] \to \mathbb R$ ($a 0$ there is no $g:[a-\epsilon,b+\epsilon] \to \mathbb R$ which is also continuously differentiable and coincides with $f$ on $[a,b]$.

I was thinking about functions which have natural definition gaps on $\mathbb R$, like $\frac{1}{x}, \log(x)$, but all of them are defined on a non compact interval.

Background: I am trying to construct a counter-example for a certain statement in real analysis.

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    My suspicion is an example may exist if you consider complex-differentiable functions on compact subsets of $\mathbb{C}$, but as stated in the answers, any continuously differentiable function defined on any compact subset of $\mathbb{R}$ (not just a compact interval) can be extended to a continuously differential equation map on all of $\mathbb{R}$. – 2017-01-23

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If $f$ is continuously differentiable on the compact interval, then it has always an extrapolation: Just take a straight line with slope $f'(a)$ at the left side and a straight line with slope $f'(b)$ at the right side of the interval. Define $g$ as the union of those lines and $f$.

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$\sqrt{x}$ is $C^1$ inside the interval [0,1] but the derivative explodes at 0.

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    Isn't it in $C^1$ in $(0,1]$? – 2017-01-23
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    I think, that we usually define $C^1$ either on an open set, or on some neighboudhood of a compact set, when we say e.g. $C^1([0,1])$. Unless we want to discuss one-sided derivatives, but then the interpolation obviously exists. – 2017-01-23
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    I thought that $C^k(A)$ was defined as the set of all points such that the $k$-th derivative does exist, other than bounded. – 2017-01-23