For those who have taken real analysis long time ago, you might have been annoyed by the same thing that has been annoying me recently during math bootcamp: How can I formally define limits and continuities? Here is my thoughts on the formal definition of limit, continuity, and uniform continuity.
Limit is defined
Continuity is defined
And uniform continuity is defined
In my opinion, the key difference between limit and continuity is how strict the definition is for each concept. In the continuity definition, we do not stipulate that as we do in the limit definition. The difference between continuity and uniform continuity is also a matter of strictness of the definition. And this difference is registered by the order of the quantifiers in our case. Uniform continuity gives us a continuous function for all in the domain, and is thus much more strict than the pointwise continuity.