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The other problem that I have is that intuitively I don't understand why uniform continuity is needed to be defined as a new concept. Now I know that pure mathematicians might be interested in a lot of abstract stuff just because they preserve some sort of structure or they allow them to prove some desirable theorems, but I think the first one who noticed that this concept is an interesting one must have had something in his mind.

My main questions are:

How should I think about uniform continuity for real functions?

Is it possible to generalize the concept of uniform continuity to more general topological spaces?

Is it possible to visualize uniform continuity in a similar fashion like how we visualize a continuous function as a function that its graph can be drawn without lifting your pen from the paper?