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I'm very excited because yesterday I asked a question about limits and I learnt that the function $f:\mathbb{Q}\longrightarrow \mathbb{R}$, such that $f(x)=x$, is continous!!, and also that generally I may define the concept of limit on functions of the kind $f:A\longrightarrow B$ where $A$ and $B$ are sets that satisfy this concept called metric. I'd like to learn more about it but I don't know where to start. Please if you could give me some orientation and tell me the requirements. Also I'll be very grateful if you could suggest some books to read. Currently I have just studied spivak's book in my first year at college.

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    $\begingroup$ The particular example you are talking about is topology, more particularly the part of topology that studies metric topological spaces. $\endgroup$ – MyUserIsThis Jul 27 '13 at 11:01
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The fields of mathematics which deal with these questions are mainly topology and functional analysis. I hope I did not forget one. Both are considered advanced subjects, but if you find a good introductory textbook to either, which lists the definitions properly and gives good examples, I would encourage you to start reading and see how far you get. In particular, if an abstract, axiomatic approach to mathematics satisfies you.

I am not experienced in topology though, and I want to check my books on functional analysis before making a recommendation

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