Countable real analysis
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916604877471744 |
|---|---|
| author | Klazar, Martin |
| author_facet | Klazar, Martin |
| contents | HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of the number $\mathrm{e}$. We also construct a uniformly continuous function $f:[0,1]\cap\mathbb{Q}\to\mathbb{R}$ such that $f'=1$ on $[0,1]\cap\mathbb{Q}$ and $\lim_{\substack{a\to1/\sqrt{2}\\a\in\mathbb{Q}}}f(a)=\frac{1}{\sqrt{2}}>f(b)$ for every $b\in[0,1]\cap\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08142 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Countable real analysis Klazar, Martin Logic Classical Analysis and ODEs History and Overview Number Theory 26A06 HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of the number $\mathrm{e}$. We also construct a uniformly continuous function $f:[0,1]\cap\mathbb{Q}\to\mathbb{R}$ such that $f'=1$ on $[0,1]\cap\mathbb{Q}$ and $\lim_{\substack{a\to1/\sqrt{2}\\a\in\mathbb{Q}}}f(a)=\frac{1}{\sqrt{2}}>f(b)$ for every $b\in[0,1]\cap\mathbb{Q}$. |
| title | Countable real analysis |
| topic | Logic Classical Analysis and ODEs History and Overview Number Theory 26A06 |
| url | https://arxiv.org/abs/2301.08142 |