The thing that has been annoying me so much: epsilon-delta proofs

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

\forall \,\varepsilon>0\,\exists\,\delta>0 \text{ s.t. } 0<|x-x_0|<\delta\,\to\,|f(x)-L|<\varepsilon

Continuity is defined

\forall x, \, \forall \varepsilon>0 \,\exists\,\delta>0\text{ s.t. }|x-x_0|<\delta\to f(x)-f(x_0)<\varepsilon

And uniform continuity is defined

\forall \varepsilon>0 \,\exists\,\delta>0 \text{ s.t }\,,\forall x, \,|x-x_0|<\delta\to f(x)-f(x_0)<\varepsilon

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 x\ne x_0 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 x in the domain, and is thus much more strict than the pointwise continuity.

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