Problem 1245 (difficulty: 4/10)

Prove that if \(\displaystyle f:\R^p\to\R\) is differentiable at \(\displaystyle a\), \(\displaystyle f(a)=0\) and \(\displaystyle f'(a)=0\), then for all bounded \(\displaystyle g:\R^p\to\R\), \(\displaystyle gf\) is differentiable at \(\displaystyle a\).

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