But teaching math absolutely is not. It is a very human process, helping students find handholds in what they know to make the next conceptual step. Absolutely breaking their minds with a new topic, then returning to it a month later and they find it obvious now. There are no platonic math lesson plans out there to tap into, and finding more effective ones has been one of humanity's jobs for at least as long as there has been written language.
(FWIW, real analysis is generally considered one of the easier courses in a typical math degree)
Quite a few colleges seem to use it as a filter class to determine who has the requisite aptitude and interest in pure math. At my college, the material wasn't particularly bad, but the evaluations were designed to be pretty challenging and grades weren't curved at all.
On the other hand, in my real analysis course we worked with metric spaces, topological spaces, Hilbert spaces, the Baire Category Theorem, etc. I doubt most students in “a typical math degree” would be studying these topics outside of a pure mathematics specific degree program.