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 thanks to this question I wrote. However, there still seems to be a crucial point that I miss conceptually that I believe/suspect will be very useful, which is distinguish clearly open maps and continuous maps. In search for this I found this question:

Open and Closed mapping Examples

which provides a non-constructive argument. But I really wanted to have a constructive argument (especially if the example was from a function from $\mathbb R$ to $\mathbb R$) of a function that is open but not continuous (I'm not interested in closed maps).

Does someone know how to construct one explicit example of it? Or at least the very least explain to me why its so hard to construct an explicit example in this case? (though that would not satisfy me as much).

I just find it unbelievable that its so hard to construct one since there has been put so much emphasis to me that they are not the same. If they are not the same then why can't we construct a simple example that just makes this fact obvious? Thats why I'm looking for hopefully, a instructive example in the simplest spaces I could think of, $\mathbb R$.

One of the main points I hope to get out of this is to understand why is it that continuous functions are able to keep nearby by points nearby while open functions do not? This is crucial for me conceptually.

• An explicit example of what, exactly? – Lee Mosher Jul 9 '18 at 18:19
• @LeeMosher sorry if that was unclear, I will add details/adjust title. I am looking for a function (hopefully from R to R or at least simple to understand) that has the property "an example of a function that is open but not continuous". My question came up due to this other question: math.stackexchange.com/questions/2052805/… – Pinocchio Jul 9 '18 at 18:58
• – Bib-lost Jul 9 '18 at 19:19
• @Bib-lost looks promising! I will take a look at it. I hope that it helps me understand why open maps are not the definition of continuity, which is what I'm really after. – Pinocchio Jul 9 '18 at 19:33
• Well, it is easy to see that an enormous amount of very continuous functions will not be open (e.g. the sine function, polynomials of even degree, constant functions, ...) – Bib-lost Jul 9 '18 at 21:17

This may not be what you are looking for, but here goes: There exists a function $f:\mathbb R\to \mathbb R$ such that $f(I)=\mathbb R$ for every interval $I$ of positive length. Such a map is clearly open, but is nowhere continuous. I build such an $f$ here: Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?
I'm not sure if this is what you want, but here it goes: every map from $\mathbb R$ into itself whose range is finite is a closed map (because every finite subset of $\mathbb R$ is closed), but such a map is continuous if and only if it is constant.
• in your example $f(X)$ does not seem to be an open set, so I am not sure how it's related to what I'm looking for actually...hmmm.... – Pinocchio Jul 9 '18 at 19:16