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.

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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.
Please clarify the doubt.


1 Answer 1


You are right. The problem statement is wrong. For example, $$f(x)=\begin{cases}(x+1)^2&x\in\Bbb Q\\0&0>x\notin \Bbb Q\\\frac1{x^2}&0<x\notin\Bbb Q\end{cases}$$ 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 $=1$.


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