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This question comes from a statement in John Milnor's "Morse Theory" on page 4.

Let $f: M \to \mathbb{R}$ be a smooth function on a manifold $M$. Milnor claims that if $a$ is not a critical value of $f$, then $M^a=\{x \in M \mid f(x) \leq a\}$ is a manifold-with-boundary, and that this "follows from the implicit function theorem."

Could someone make explicit to which function we're applying the implicit function theorem, and how the above statement follows?

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Hint: We need to exhibit a suitable atlas. Recall what "is not a critical value" means. Infer from that which Jacobian might be invertible and from that get an idea how the IFT is applied.

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