I want to know if this statement is true.
Let $F: M \to N$ a function between manifolds, such that the restriction of $F$ to each element of an open cover $\mathcal{O}$ of $M$ is differentiable. Is $F$ a differentiable function?
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Differentiability is a local property; differentiability at a point cares only about what happens on incredibly small open neighbourhoods around that point. So I don't see why not. There might be some subtlety here I'm missing, though. – Arthur Jan 15 '17 at 01:16
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@Arthur - you are not missing anything. This is practically the definition of $F$ being differentiable (if $\mathcal O$ consisted of coordinate maps, then it would exactly be the definition). – Paul Sinclair Jan 15 '17 at 04:38