Product with funcion non-differentiable at one point.

Question

Suppose that $U$ is an open set containing $0.$ $f,g:U\to\mathbb{R}$ are continuos functions such that

• $g\in C^1(U)$
• $g(0)=0$
• $f|_{U\setminus 0}\in C^1(U\setminus 0)$ ($f$ is of $C^1$ class away of $0$)

Question. Does product $fg$ is of $C^1$ class, i.e $fg\in C^1(U)$?

I can show that $fg$ is differentiable at $0.$ However I am not sure if derivative is continuous at $0.$

Answer 1

No. Let $g(x)=x$ and $f(x)=x \sin(1/x)$ for $x \ne 0$ and $f(0)=0$

Then $(fg)(x)=x^2 \sin(1/x)$ for $x \ne 0$ and $(fg)(0)=0$.

Show: the derivative of $fg$ is not continuous at $x=0$

Answer 2

Hint: The function $$ h(x) = \begin{cases} x^{2}\sin(1/x) & x \neq 0, \\ 0 & x = 0, \end{cases} $$ is well-known (and easily seen) to be differentiable, but not $C^{1}$ at $0$. Factorizations are easily found that give counterexamples of the type you seek.