Suppose $\zeta: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a $C^1$ function that is total differentiable with $\zeta (0,0) = (0,0)$ and where $(d\zeta)(0,0)$ is invertible. Write $\zeta_1$ as the first component function.
Show that for any function $f: \mathbb{R} \rightarrow \mathbb{R}$ it holds that if $f \circ \zeta_1$ is total differentiable in $(0,0)$ then $f$ is differentiable in $0$.
This was part of an exam for an introductory course of real analysis. I wasn't able to find the solution in time, but I haven't had time to revisit the problem until now.