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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.
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
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
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
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
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 …
- 4,713