I was wondering if there exists a book that talks to highschool level audience who wish to study pure mathematics for their undergraduate years. I'm not talking about problem solving kind of books (I've found a lot, since math competitions are very famous). But more like a book that talks of what to expect in, for example, calculus or algebra.
The book "How to think like a mathematician" by Kevin Houston. It was recommended reading during the summer before I enrolled in my undergraduate math program, and I found it incredibly helpful. It discusses how math at university level differs from math in high school. You get a nice introduction to what mathematicians mean by definitions, lemmas, theorems, and a number of different proof techniques. It's written specifically to help bridge the gap between high school and university, but it can also help when you want to see what math at university is like.
What is Mathematics by Courant and Robbins is what I usually recommend to my highschoolers, closely followed by The Mathematical Experience by Davis and Hersh. Both of these would seem to fit the bill quite neatly - a good overview of topics like topology, abstract algebra, real analysis etc with some nice problems to work through (although not strictly necessary).
I find the books by William Dunham ("Euler: Master of Us All", "Journey Throug Genius", "The Mathematical Universe", there are others) very inspiring. Mostly at high-school level, but they require quite a bit of effort to really grasp the details. Aigner and Ziegler's "Proofs from THE BOOK" is a bit harder, but shows off many dazzling proofs. None are really about "what is math at college like", but they do show a bit of the history and some gems.