Smoothness along rays sufficient for global smoothness - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:40:36Z http://mathoverflow.net/feeds/question/73551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73551/smoothness-along-rays-sufficient-for-global-smoothness Smoothness along rays sufficient for global smoothness Bart 2011-08-24T09:20:48Z 2011-08-24T09:20:48Z <p>Hi,</p> <p>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 $d \in \mathbb{R}^d$.</p> <p>Is smoothness along these rays sufficient for $f$ to be smooth around $0$ as a multivariate function (all partial derivatives exist)?</p> <p>Thanks.</p>