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Results tagged with real-analysis
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user 8250
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
3
votes
A net of lower semicontinuous functions
Let us denote $I=[0,1]$ and let us choose $\varepsilon_n=2^{-n}$.
Notice that in the situation in the question we have $f=\sup\limits_{\alpha\in A} f_\alpha$, and thus $f$ is a lsc function.$\newcomma …
- 4,713
4
votes
Is there a Borel-measurable function which maps every interval onto $\mathbb R$?
This is Exercise 9.M from A. C. M. van Rooij, W. H. Schikhof: A Second Course on Real Analysis.$\newcommand{\dcc}[1]{\lfloor#1\rfloor}$
Exercise 9.M. (Another function that maps every interval ont …
- 4,713
2
votes
Accepted
Sets closed by sum and solutions to the Cauchy functional equation
Suppose we have a decomposition $\mathbb R=A\cup B$ as described in the question and that $0\in A$. (Clearly, $0\in B$ is impossible, so we have to denote by $A$ the set which contains zero.)
Now, le …
- 4,713
31
votes
Can the supremum of continuous functions be discontinuous at every point of an interval?
Since the function $f$ is supremum of a set of continuous functions, it is lower-semicontinuous.1
Every lower semicontinuous function belongs to the first Baire class.2
If $f\colon \mathbb R\to\math …
- 4,713
4
votes
Articles with examples of Darboux functions without fixed points
In Exercise 9.C of van Rooij - Schikhoff: A Second Course on Real Functions the following example is given.
Take any function $f \colon \mathbb R\to\mathbb R$ such that $f[I]=\mathbb R$ for every …
- 4,713
4
votes
Accepted
Meeting a set of lines in $\mathbb{R}^n$
If we have a set $\mathcal L$ of lines in $\mathbb R^n$ such that $|\mathcal L|=\mathfrak c$, we can get the set $M$ with the desired properties using transfinite induction.
Take any well-ordering o …
- 4,713
4
votes
Accepted
Do monotone functions on the interval have an "Alexander duality" property?
Let us try to construct $S$ which is a counterexample to (B) by transfinite induction. (In the style of just-do-it1 proofs.)
We will use the fact2 that the set $\mathcal M$ of all monotone non-decrea …
- 12.9k