Find the fallacy in the following proof

Question

This is in reference to this question:Derivable product of derivable and non-derivable function, not satisfying product rule

I gave a proof which is given here https://math.stackexchange.com/a/2926420/595861 and at the same time a counter-example has also been provided here https://math.stackexchange.com/a/2926425/595861

I don't understand how is this possible.Can someone kindly help me to find the fallacy in the proof?

Answer 1

The problem lies in the sentence “Since $\lim_{x\to 0}\dfrac{f(x)-f(0)}{x}$ exists and $g(0)=0$ so $\lim_{x\to 0}\dfrac{g(x)\{f(x)-f(0)\}}{x}=0$.” This would be correct if we were assuming that $g$ is continuous at $0$, but we aren't.

Answer 2

You may be using that $g(x) \to 0$ when $x \to 0$. This is not true if $g$ is not continuous. Note that the counterexample given exploits this precise fact, taking

$$ g(x) = \cases{\frac{1}{x} \quad x \neq 0 \\ 0 \quad \text{otherwise}} $$