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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 = \...
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### 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 ...
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### 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 $(0,1)$ ?
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### 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 ...
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### 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 ...
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### 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 ...
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 ...