Definition:Differentiable Mapping between Manifolds

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Definition
Let $M$ and $N$ be differentiable manifolds.

Let $f : M \to N$ be continuous.

Definition 1
$f$ is differentiable  for every pair of charts $(U, \phi)$ and $(V,\psi)$ of $M$ and $N$:
 * $\psi\circ f\circ \phi^{-1} : \phi ( U \cap f^{-1}(V)) \to \psi(V)$

is differentiable.

Definition 2
$f$ is differentiable  $f$ is  differentiable at every point of $M$.

Also see

 * Definition:Smooth Mapping Between Manifolds