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Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point on the boundary $\partial F(M)$ is a critical value. Of course it is also possible that a critical value is contained in the interior.

Is there any way of distinguishing critical points that are mapped to the boundary of the image from critical points that are mapped to the interior?

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  • $\begingroup$ This is obviously not a local property. $\endgroup$
    – Alex Degtyarev
    Commented May 6, 2016 at 17:24
  • $\begingroup$ @AlexDegtyarev I agree that a complete (if and only if) classification cannot be purely local, but still think a partial answer may be possible. For instance, in the one-dimensional case the boundary of $F(M) \subset \mathbb{R}$ just consists of the maximum and minimum values of $F$, so any critical point that is not a local extrema must be mapped to the interior. $\endgroup$
    – Graham Cox
    Commented May 7, 2016 at 2:06

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