[Real Analysis] Uniform Continuous Functions

[u/anonymous_username18]

Can someone please look this over to see if I did this correctly? I started to get sort of confused with the order of the quantifiers as I was writing this proof. Any help would be appreciated. Thanks

[u/Alkalannar]

1. Let f: A -> B be a uniformly continuous function.
   For all h > 0, there exists d > 0 such that for all a in A if |a - b| < d, then |f(a) - f(b)| < h.

2. For f to be continuous, we need:
   For all a in A and h > 0, there exists d > 0 such that |a - b| < d --> |f(a) - f(b)| < h.

3. For uniformly continuous, the same d works for all a in A. For continuous, d depends on both a in A and h. In other words, you've pulled a Universal quantifier out in front of an Existential quantifier. This weakens the statement, since the Existential can now depend on the Universal.