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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.
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
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