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?
Hint: You just need to write it down clearly. $c$ is smooth requires you to show that $\phi \circ c \circ \psi^-$ is smooth. Now as $c$ is constant,for all arguments, you are left with $\phi(y)$ in the end. Now this is just a constant. Use definition of smoothness now