Linked Questions

8
votes
3answers
3k views

Function whose image of every open interval is $(-\infty,\infty)$ [duplicate]

How to find a function from reals to reals such that the image of every open interval is the whole of R? Is there one which maps rationals to rationals?
2
votes
1answer
64 views

Do these functions exist? [duplicate]

I created this question, but, I do not know the answer: Is there a function $f: \mathbb R \to \mathbb R$ such that for every interval $I \subseteq \mathbb R$ we have $f(I)=\mathbb R$? It seems to ...
0
votes
1answer
46 views

Question about the existence of a specific function $f: \mathbb{R} \to \mathbb{R}$ [duplicate]

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all non-empty, open, and uncountable subset $E \subseteq \mathbb{R}$, $f(E) = \mathbb{R}$? I have been thinking about it for a ...
36
votes
4answers
2k views

Why weren't continuous functions defined as Darboux functions?

When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like "Continuous functions are those you can draw without lifting your pen" With this in ...
29
votes
5answers
1k views

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? I think it is probable because we can consider $$ y = \...
39
votes
3answers
7k views

Open maps which are not continuous

What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of ...
7
votes
1answer
205 views

A map from $(0,1)$ to $(0,1)$ such that the image of every open interval in $(0,1)$ is $(0,1)$

Can we have a map from $(0,1)$ to $(0,1)$ such that the image of every open interval in $(0,1)$ is all of $(0,1)$ ?
2
votes
3answers
128 views

Show that there exists a real function of the real variable that takes any real value on any interval

Show that there exists a functon $f:\mathbb R\to\mathbb R$ such that if $I$ is any nonempty open interval then $f(I)=\mathbb R$. For now I do not see how to approach this exercise. Would you have a ...
1
vote
2answers
86 views

Explicit/constructive example of open maps that are not continuous (especially from R to R)?

TLDR: I'm looking for an explicit map that is an open map but not continuous. The context my question arose was when learning the topological definition of continuous function. I made some progress ...
0
votes
2answers
51 views

Every circle passes through points of all colors

Let $n$ be a positive integer. Is it possible to color every point in the plane in one of $n$ colors so that every (nondegenerate) circle contains points of every color? If we can do the coloring so ...
1
vote
1answer
56 views

Can connectedness preservation be used to define continuity of a function?

Suppose we have a function from a topological space (X, Tx) to (Y, Ty). If for each subset of X that is connected (with respect to the induced topology from X) the image of that subset is also ...