Lemma about smooth functions on manifolds with boundary

Question

Let $f:M\to N$ be a continuous map between smooth manifolds with boundary. Is it true that smoothness of both $f|_{\mathring{M}}:\mathring{M}\to N$ and $f|_{\partial M}:\partial M\to N$ implies that $f$ is smooth? I would say so, but I am unable to start a proof. Thanks in advance for any hint!

Answer 1

No. If $M=[0,\infty)$, $N=\mathbf{R}$ and $f(x)=\sqrt{x}$, then $f$ is smooth on $\mathring{M}=(0,\infty)$ and on $\partial M=\{0\}$, but not on $M$.