Question: Let $M$ , $N$, and $P$ be smooth manifolds with or without boundary. 

Every constant map $c: M\rightarrow N$ is smooth.

**Proof:** Let $c: M \rightarrow N$ be a constant map. Let $p \in M$. Smoothness of $c$ means there are charts $(U,\phi)$ of $p$ and $(V,\psi)$ of $c(p)$ such that $c(U) \subseteq V$ and $\psi \circ c \ \circ \phi^{-1} $ is smooth. Since $c$ is a constant map we know that $c(p)=y$ for every $p \in M$. 

This is as far as I got with the proof. I'm a bit lost on how to finish the proof using the fact that c is a constant map to show that $c: M \rightarrow N$ is smooth.

I'd appreciate hints or advice instead of a full solution to the problem that way it doesn't spoil the problem for me. 

I'm using Lee's Introduction to Smooth Manifolds.

Answer: 
We have chosen some chart $\phi \colon U \rightarrow \phi(U) \subseteq \mathbb{R}^n$ on $M$ and a chart $\psi \colon V \rightarrow \psi(V) \subseteq \mathbb{R}^m$ on $N$. We want to show that $\psi \circ c \circ \phi^{-1}$ is smooth given $c$ is constant. 

Hints: 1. What is the domain and codomain of the map $\psi \circ c \circ \phi^{-1}$

2. Is a constant map $k \colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ smooth?
***Every constant map is smooth.***

> 
> Say $f:M \rightarrow N $ is a constant map. Here $f$ is smooth because the coaordinate representation of $f$ i.e $\theta f\phi^{-1} :R^{m}\rightarrow R^n$ is a constant map as a map between 2 euclidean spaces where $(u,\phi)$ and $(v,\theta)$ are charts around $p$ and $q$ respectively where $f(p)=q$ $\forall$ $p \in$ $M$. Here dimension of $M$ and $N$ are assumed to be $m$ and $n$ respectively. We already know from multi variable calculus that any constant map between two euclidean spaces is always smooth. Hence by definiton of a smooth map between smooth manifolds $f$ is smooth.
> 
> 
>