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Is there a compactly supported and smooth (except at one point) function $f : \mathbb{R} \to \mathbb{R}$ where (non smooth point) it is is exactly k times differentiable ?

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Yes. Just take $f(x)=0$ for $x<0$, $f(x)=x^{k+1}$ for $x \ge 0$, and multiply by a smooth transition function (say one that is $1$ for $x<1$ and $0$ for x>2) to make it compactly supported.

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    @Willie Wong : thank you2011-01-18