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Results tagged with real-analysis
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user 109573
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
4
votes
Sequential continuity and the Axiom of Choice
If the graph of $f$ has a Borel code $c,$ then ZF proves equivalence of these two continuity concepts. Discontinuity of $f$ at $x$ is downward absolute to $L[c, x],$ in which we can find a sequence $x …
- 4,316
13
votes
Is it consistent with ZFC that the real line is approachable by sets with no accumulation po...
Here's a ZF proof that if $S$ is a chain of sets with $\bigcup S = \mathbb{R},$ then there is $X \in S$ which contains a countable set dense in some nonempty open set.
If there is $X \in S$ such that …
- 4,316
5
votes
Accepted
The difference between Baire 2 and 'effectively Baire 2'
It's provable in ZF that every Baire-2 function is effectively Baire-2. It suffices to prove the following:
(ZF) There is an explicit function which maps each Baire-1 function $f: \mathbb{R} \rightarr …
- 4,316