real analysis - Is there any function in this way?

$f$ is a function which is continous on $\Bbb R$, and $f^2$ is differentiable at $x=0$. Suppose $f(0)=1$. Must $f$ be differentiable at $0$?

I may feel it is not necessarily for $f$ to be differentiable at $x=0$ though $f^2$ is. But I cannot find a counterexample to disprove this. Anyone has an example?

Answer

Differentiability of $f$ is existence of $\lim(f(x)-1)/x$. Differentiability of $f^2$ is existence of $\lim((f(x))^2-1)/x$. Consider factoring the numerator of the latter and thinking about the consequences.