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11:10 AM
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Q: Continuity and the Axiom of Choice

Donkey_2009In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies $f(z_n) \to f(a)$. $(2)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if $\forall ...

I like the fact that:
a) We need (some form of) AC to prove that the two definitions of continuity at a point are equivalent.
b) If we consider global continuity, then the two definition are equivalent in ZF. Although the proof is more complicated. (This result is due to Sierpniski.)
 

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