I suspect, for a lot of math students, Real Analysis is the first introduction to "real math" and is frequently taught assuming such. I found, at least in undergrad, that a fair bit of students that think they are "good at math" are really "good at calculation" and Real Analysis is quite a shock for anyone coming from that perspective.
It's unfortunate that many students studying math more causally or as a prerequisite for other fields don't get a chance to study Real Analysis because, in addition to this difficulty (from lack of exposure), it's also a great introduction to the beauty of mathematics.
I'm about as far as one can get from a practicing mathematician, but I still find myself pulling out the baby Rudin from time to time just for the pure pleasure of wandering through it.
I had baby Rudin for my undergrad class (I don't remember which book I had in grad school), and I kept that one when I downsized my book collection in case I want to refresh my memory someday. At the time, I was more interested in algebra and combinatorics. I also kept my copy of Herstein.
25 years later I've forgotten most of it. But recently I've turned my attention to type theory, category theory, and related topics. I have little spare time and the amount of interesting topics to explore is daunting, but it's fascinating. (I've always been interested in mathematical foundations, too.)
It's unfortunate that many students studying math more causally or as a prerequisite for other fields don't get a chance to study Real Analysis because, in addition to this difficulty (from lack of exposure), it's also a great introduction to the beauty of mathematics.
I'm about as far as one can get from a practicing mathematician, but I still find myself pulling out the baby Rudin from time to time just for the pure pleasure of wandering through it.