The "Heine–Cantor theorem" states: If $f : M → N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous.
I do not doubt its validity, of course, just trying to understand why it is valid.
If we, say, take the function: $y = x^4$. It rises very quickly with the rising value of the argument. How is it that according to Heine–Cantor theorem just because we enclose the argument of the function, say, between $[0, 10]$ it automatically becomes "uniformly continuous" (considering of course that it is "continuous")?
There are areas of function values inside the argument segment where function will rise quicker than in some other areas.
Does the reason have to do with the fact that we could in worst case choose $\delta=10$ (length of the segment in example) and thus cover all possible cases for $\varepsilon$?