Let $f:U\to V$ be a continuous map of open subsets of Euclidean spaces $U\subset\mathbb R^m,V\subset\mathbb R^n$. Suppose:
- For every differentiable curve $\gamma$ in $U$ based at $p\in U$ the composite $f\circ \gamma$ is a differentiable curve in $V$ based at $fp$.
- There's a well-defined assignment $\gamma^\prime(0)\mapsto (f\circ \gamma)^\prime(0)$ which is moreover a linear map $\mathbb R^m=\mathrm T_pU\to \mathrm T_{fp}V=\mathbb R^n$.
Does it follow that $f$ is differentiable at $p$?
Example 3.3 of Kriegl and Michor's The Convenient Setting of Global Analysis provides a non-differentiable function $\mathbb R^2\to \mathbb R^2$ satisfying the first condition, but it does not seem to satisfy the second.
