Intereting Posts

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Hahn-Banach theorem: 2 versions
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Squarefree binomial coefficients.
Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion
How to calculate the following limit: $\lim_{x\to+\infty} \sqrt{n}(\sqrt{x}-1)$?
Manipulation of Bell Polynomials
Split up sum of products $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ for uncorrelated summands?
the knot surgery – from a $6^3_2$ knot to a $3_1$ trefoil knot
Calculate the multiplicative inverse modulo a composite number
Books for a beginner
If $f$ is strictly convex in a convex set, show it has no more than 1 minimum
Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$.

(Take $U_n = \bigcup_{y \in Im(f)} f^{-1}(B_{\frac{1}{n}}(y))$ and $V= \bigcap_{n \in \mathbb{N}}U_n$, and $V$ is $G_\delta$).

My question is: Is the converse direction true? Is it true that, given a $G_\delta$ set $A \subseteq X$, there exists a metric space $Y$, and a function $f:X \rightarrow Y$, such that, the set of points on which $f$ is continuous is $A$?

- How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$
- Does a nonlinear additive function on R imply a Hamel basis of R?
- Prime Harmonic Series $\sum\limits_{p\in\mathbb P}\frac1p$
- a Fourier transform (sinc)
- A difficult integral evaluation problem
- Projection of a set $G_\delta$

Thank you!

Shir

- Is metric (Cauchy) completeness “outside the realm” of first order logic?
- Negation of uniform convergence
- Showing the exponential and logarithmic functions are unique in satisfying their properties
- Taylor series for different points… how do they look?
- Let $f:(\mathbb{R}\setminus\mathbb{Q})\cap \to \mathbb{Q}\cap $. Prove there exists a continuous$f$.
- An example of topological space in which each singleton is not in $G_\delta$
- Limit of solution of linear system of ODEs as $t\to \infty$
- Discontinuous functions that are continuous on every line in $\bf R^2$
- Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
- Is my proof of the uniqueness of $0$ non-circular?

Yes. See Theorem 2.1 on page 2 of

http://artsci.kyushu-u.ac.jp/~ssaito/eng/maths/Gdelta.pdf

In fact, every dense $G_\delta\subset\mathbb R$ is also the set of continuity of a derivative of a differentiable function!

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