# Definition:Differentiable Mapping between Manifolds/Point

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## Definition

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

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

Let $p \in M$.

#### Definition 1

$f$ is **differentiable at $p$** if and only if for every pair of charts $\struct {U, \phi}$ and $\struct {V, \psi}$ of $M$ and $N$ with $p \in U$ and $\map f p \in V$:

- $\psi \circ f \circ \phi^{-1}: \map \phi {U \cap \map {f^{-1} } V} \to \map \psi V$

is differentiable at $\map \phi p$.

#### Definition 2

$f$ is **differentiable at $p$** if and only if there exists a pair of charts $\struct {U, \phi}$ and $\struct {V, \psi}$ of $M$ and $N$ with $p \in U$ and $\map f p \in V$ such that:

- $\psi \circ f \circ \phi^{-1}: \map \phi {U \cap \map {f^{-1} } V} \to \map \psi V$

is differentiable at $\map \phi p$.