If the total derivative $Df(x)$ of $f:\mathbb{R}^n \to \mathbb{R}^m$ is continuous does it imply that all partial derivatives are continuous?
I would say yes of course because one can simply take the composition $(f \circ\pi_i)(x_i)$, where $\pi_i: \mathbb{R} \to \mathbb{R}^n$ and $\pi_i(x_i)=(a_1, ..., x_i, a_{i+1},..., a_n)$ for a point $a=(a_1, ..., a_n)\in \mathbb{R}^n$. As both functions $\pi_i$ and $Df(x)$ are continuous their composition is also continuous. But $Df(\pi_i(x_i))$ yields exactly the $i$-th partial derivative.
Is this correct?