Is there a theorem which guarantees me that any function $f$ with bounded first and second order derivatives defined over a compact interval of $\mathbb{R}^2$ can be extended to a twice continuously differentiable function $F$ on the whole space $\mathbb{R}^3$?
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I think your condition is not enough. Since initially you only require the function have bounded second derivative on a compact subset, since differentiable doesn't imply continuous differentiable, it may not be continuous differentiable on the compact set, so it can't be extended to $\mathbb{R}^3$.
An example is $$f(x) = x^{4} \cdot \sin\left(\frac{1}{x}\right)$$ $$f(0) = 0$$
The $2$nd derivative of $f(x)$ exists for each value of $x$, bounded at the neighborhood of $0$ but not continuous at $x = 0$.
John
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Good point! Thank you. If you add continously differentiable, does there then exist such a theorem? (I dont know if this is somehow important, but in my particular case I have a symmetric function f(x,y)) – HPWeight Nov 21 '14 at 20:08