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:
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.