Hi,

Suppose I have a function $f:\mathbb{R}^d \to \mathbb{R}$ and I know that $f$ is smooth ($C^\infty$) along each ray $t \mapsto f(td)$ on $t \in [-\epsilon, \epsilon]$ and all directions $d \in \mathbb{R}^d$.

Is smoothness along these rays sufficient for $f$ to be smooth around $0$ as a multivariate function (all partial derivatives exist)?

Thanks.