If $f$ is a smooth function on a smooth manifold $M$ , then $df = 0$ if and only if $f$ is constant on each component of $M$

Question

I am reading introduction to smooth manifolds by John M. Lee.

Proposition 4.9

If $f$ is a smooth function on a smooth manifold $M$ , then $df = 0$ if and only if $f$ is constant on each component of $M$

And proof starts:

"It suffices to assume that $M$ is connected and show that $df = 0$ if and only if $f$ is constant."

Why we can assume that $M$ is connected?

In the middle of the proof there is a sentence:

"If Let U be a connected coordinate domain centered at $q$."

What is connected coordinate domain?