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Given $X$ is compact and $Y$ connected, and $f$ is a submersion.

How to draw the conclusion that $f$ is continuous?

In my book, submersion is defined as:

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WishingFish
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    Usually the term "submersion" only refers to a map that is $C^k$ for some $k$, and hence already continuous. What is your definition of "submersion"? – Zev Chonoles Jul 05 '13 at 19:36
  • Thanks a lot @ZevChonoles, please see the update. =) – WishingFish Jul 05 '13 at 19:40
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    That doesn't really address my concern - how is the derivative defined if the map $f$ is not at least $C^1$, and hence already continuous? – Zev Chonoles Jul 05 '13 at 19:43
  • May I grant the assumption that $f$ is $\mathcal{C}^1$ continuous since it is talking about first-order derivative? – WishingFish Jul 05 '13 at 19:44
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    That is entirely reasonable to me - but that then raises the issue of what the question could possibly be intended to mean? – Zev Chonoles Jul 05 '13 at 19:45
  • Yeah, I know the definition does not address your concern. But I tried to dig out the information that $\mathcal{C}^k$ differentiable, and the definition is what I found out.. – WishingFish Jul 05 '13 at 19:45
  • Oh, I was not sure if I can grant $\mathcal{C}^1$ differentiable. Thanks @Zev. – WishingFish Jul 05 '13 at 19:46

1 Answers1

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A function that is differentiable at $x$ is continuous at $x$ (proved in Euclidean case in any textbook on real analysis; the extension to manifolds uses the fact that chart maps are homeomorphisms).

Therefore, a function for which $df_x$ exists at every point is continuous at every point. The surjectivity of $df_x$ is not needed to obtain the continuity of $f$.

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