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18 votes
0 answers
1k views

Does there exist a continuous open map from the closed annulus to the closed disk?

(Originally from MSE, but crossposted here upon suggestion from the comments) In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
D.R.'s user avatar
  • 831
0 votes
1 answer
969 views

Is the pointwise supremum of a continuous function continuous?

Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
Saeed's user avatar
  • 3
3 votes
2 answers
516 views

Several definitions of approximate continuity of real functions

I found the definition of approximate continuity stated as follows: A function $f:\mathbb R \to \mathbb R$ is approximately continuous at $x_0$ iff there exists a set $A\in \mathcal L$ such that $x_0\...
MAS's user avatar
  • 930
0 votes
1 answer
282 views

Continuity of the Restriction Map Between Function Spaces [closed]

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as \begin{align} \rho:&C(X,Y)\rightarrow C(Z,...
ABIM's user avatar
  • 5,405
21 votes
3 answers
610 views

Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
Trevor J Richards's user avatar
7 votes
0 answers
227 views

Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
Taras Banakh's user avatar
  • 41.8k
7 votes
1 answer
2k views

If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?

Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
Silvio Levy's user avatar