Probably at more elementary level than you intend, but otherwise (I think) remarkably on target,

- <cite authors="Burn, R. P.">_Burn, R. P._, Numbers and functions: steps into analysis., Cambridge: Cambridge Univ. Press. xix, 328 p. (1993). [ZBL0872.00009](https://zbmath.org/?q=an:0872.00009).</cite>

(I wish I remembered where I recently saw it recommended. [This related question](https://mathoverflow.net/questions/12709/are-there-any-books-that-take-a-theorems-as-problems-approach) mentions it along with others by the same author.)