Let $f: \mathbb{R}^m\rightarrow \mathbb{R}$ be an everywhere continuous function and suppose there exists a linear map $L: \mathbb{R}^m\rightarrow \mathbb{R}$ such that $$\lim_{t\rightarrow 0} \frac{f(tv)-f(0)}{t}= L(v)$$ for every unit vector $v\in \mathbb{R}^m$. Is f differentiable at 0?
This is motivated by a proof of the fact that, for a Lie group G, $f\in C_c^\infty(G)$ implies that $f$ is a smooth vector for the regular representation of $G$ on $L^2(G)$. The author claims that it suffices to show the directional derivatives at $e\in G$ are given by the differential $X \mapsto Xf$, where $X\in \frak{g}$ is thought of as a differential operator.