# To prove differentiability of a function at a point

I am studying Denlinger's Elements of Real Analysis, there a problem is given as-

Suppose $$f:\mathbb{R}\to\mathbb{R}$$ is differentiable at $$x_o$$, and define $$g:\mathbb{R}\to\mathbb{R}$$ by $$g(x)=x^3f(x^2)$$. Prove that g is differentiable at $$x_o$$ and find $$g'(x_o)$$.

I have attached the pic of my work.

I have a doubt that we are just given that f is differentiable at $$x_o$$. Please tell is my way incorrect or I am missimg something.
You are right. The problem statement is wrong. For example, $$f(x)=\begin{cases}(x+1)^2&x\in\Bbb Q\\0&0>x\notin \Bbb Q\\\frac1{x^2}&0 is differentiable at $$x_0=-1$$, but $$g(x)=x^3f(x^2)$$ is nowhere differentiable (in fact, nowhere continuous).
The problem statement should ask for differentiability at $$\sqrt{x_0}$$ (hoping that $$x_0\ge0$$) or specify $$x_0=0$$ or $$=1$$.