Calculus or Analysis?

My first attempt at teaching an introductory course in real analysis went well enough I thought. The students came to understand the logical structure of the proper definition of a limit and we used it to prove that polynomials really are continuous. I introduced enough topology of the real line to show that continuous functions on compact sets are uniformly continuous and attain extreme values, and then pressed on to show how this leads to an elegant proof of the Mean Value Theorem for the derivative. In the last part of the term we made a proper pass through the theory of the Riemann integral and, as a big finish, used our rigorously justified Mean Value Theorem to construct an argument for the Fundamental Theorem of Calculus. When the dust settled there was plenty to be proud of. The course evaluations were generally positive, the students learned how to write a proper ∈–δ proof and, as far as I could tell, no one had gotten hurt along the way.

Although it took several years of thinking and tinkering before I was able to put my finger squarely on why my first versions of this course felt oddly unsatisfying, the conclusions I reached are hardly revolutionary.