$\vert f \vert$ is always differentiable at $x=0$

Question

Let $f:(-1, 1) \rightarrow \mathbb{R}$ be a function satisfying

$$ \frac{1}{2}\vert x \log \vert x \vert \vert \leq \vert f(x) \vert \leq\vert x \log \vert x \vert \vert$$

for all $x \neq 0.$ Then

1) $f$ is never differentiable at $x=0$

2) $f$ is always differentiable at $x=0$

3) $f$ is differentiable at $x=0$ if $f(0)=0$

4) $\vert f \vert$ is always differentiable at $x=0$

How doi I approach? I tried using using the definition at $x=0$ , Can I say $f(0)=0$ from the given inequality using squeeze lemma. Please help. Thanks.

Answer 1

The correct answer is $1$. Let us assume that $f$ is differentiable in $0$. Then $(f(x)-f(0))/x$ has a finite limit for $x$ going to $0$. If $f(0)\neq 0$ then $\frac{|f(x)|}{|x|}< |\log(|x|)|=o(\frac{f(0)}{x})$ when $x$ goes to $0$ and the expression diverges. Impossible.

So $f(0)=0$. But then $\frac{|f(x)|}{|x|}>\frac{1}{2} |\log(|x|)| \to_{x \to 0} +\infty$ diverges. This contradicts the hypothesis and so $f$ is never differentiable in $0$.