Why is $M_{f\geq 0}$ a manifold with boundary?

Question

I'm studying Morse theory and I found this fact:

Let be $f:M\rightarrow \mathbb{R} $ a Morse function on a m-manifold $M$. Suppose $0$ is not a critical value of $f$. Then $M_{f\geq 0}=\lbrace p\in M | f(p)\geq 0\rbrace$ is a manifold with boundary and $\partial M = M_{f=0} = \lbrace p\in M | f(p)= 0\rbrace$.

I have problems with this. I know that points in $M_{f=0}$ must not be interior points, but the other points in $M_{f\geq 0}$ are interior points. However, I don't know why points in $M_{f=0}$ are boundary points...

Answer 1

Suppose $p$ is an interior point of the submanifold $M_{f\ge0}$. Pick a chart around $p$ so that we're working with a function $f : \mathbf{R}^m \to \mathbf{R}$ having the following properties:

1. $(df)_0 \neq 0$
2. $f(0) = 0$
3. $f(x) \ge 0$ for all $x$

Conditions 2 and 3 say that $0$ is a local minimum (a global minimum on a local chart) so how could the derivative be non-zero?