Suppose $M$ is a smooth manifold and $f:M\rightarrow\mathbb{R}$ a Morse function. Let $$M_a := \{p\in M\,|\,f(p)<a\}.$$ Then $\overline{M_a} = \{p\in M\,|\,f(p)\leq a\}$ and $\partial\overline{M_a} = \{p\in M\,|\,f(p) = a\} = f^{-1}(\{a\})$.
Is it true that $\overline{M_a}$ is a smooth manifold with smooth boundary $\partial\overline{M_a}$ if and only if $a$ is a regular value? How does one see this?