> [!theorem] Morphisms are continuous > > Let $X, Y$ be $C^p$ [[Manifold|manifolds]] and $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], then $f$ is [[Continuity|continuous]]. > > *Proof*. Let $p \in X$, $(U, \psi) \in X$, and $(V, \varphi) \in Y$ such that $p \in U$ and $f(U) \subset V$. Let $N \in \cn(f(p))$ be any neighbourhood. Since $V$ is open, assume without loss of generality that $N \subset V$. Since $f|_U = \psi^{-1} \circ f_{U, V} \circ \varphi$ is a composition of continuous maps, it is continuous, and > $ > U \cap f^{-1}(W) = (f|_U)^{-1}(W) > $ > is open in $X$. Therefore the preimage of neighbourhoods are neighbourhoods, and $f$ is continuous.