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I suppose the canonical example is the function which maps n to 1 if n is the GN of a TM which halts on a blank tape and n to 0 if n is the GN of a TM which never halts on a blank tape. This is not recursive, because if it were we would be able to solve the BTHP. This seems easily described to me. You may need to define "arithmetical"...
I am having exactly this issue currently. I live in Australia, and as a result there are not that many mathematics courses offered at my university (in particular, there are no graduate courses). In order to graduate i have had to take many grunt courses
@Andrew L: I had a look in Pugh's book this afternoon. I am inclined to agree with you. The taste of topology chapter in particular looks miles better than its counterpart in Rudin.
@ Andrew L: Rudin was my first introduction to analysis. In hindsight, i knew what a derivative was (but not to much more) and i could mechanically evaluate integrals from high school. Max seems to know a lot more than I did when I first picked up rudin, and I didn't have to much trouble working my way through it. I am not telling him that he should read rudin, I am just telling him that I think he would be able to handle it if he wanted to and I think it is a good book.
I don't think Max here is an ordinary student. He seems like he is interested in becoming a mathematician, and he seems very bright. In this light, even if he is not able to handle Lee's topological manifolds straight away, I would be very surprised if he cant handle Rudin's Principles of Mathematical analysis. Once he has mastered this book, he should be able to move onto Lee's book easily.
