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

Question

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:

Answer 1

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 $f$.