#diffgeo 

## Definition

Let $M, N$ be two [[Manifold]]s with maximal [[Atlas]]es $A_M, A_N$. A map $f: M \to N$ is a $C^r$-*map*/$C^r$-*function* in $p \in M$, if maps $(V, y) \in A_N$ with $f(p) \in V$ and $(U,x) \in A_M$ with $p \in U$ exist, such that $y \circ f \circ x^{-1}$ (a function between open subsets of $\mathbb{R}^n$, $\mathbb{R}^m$) is in $C^r$.

If $r = \infty$, then $f$ is *smooth*.