World's elite universities will tell you right away that as their CS student you are considered the best group they have and that math students go slower than you are, and increase your load to crazy levels. I am not going to post "unsubstantive" comments that I haven't heard at those places. As a CS student, you are expected to master (continuous) calculus, discrete calculus (discrete math proofs, hypercubes for parallel algorithms), optimization (machine/deep learning, compilers), category theory (functional programming), logic (up to automated proofs, i.e. including set theory), differential equations, topology (computational geometry, distributed algorithms), probability and statistics (reinforcement learning, queueing), number theory (cryptology), graph theory (almost everywhere)... There is no functional analysis needed yet, but it's heavily used for PhD degrees anyway. You need to know all this down to the level of proving theorems if you want to achieve anything in CS.
I know several CS PhDs from “world elite universities” with undergraduate math or physics degrees who switched to CS for grad school because they decided that math/theoretical physics were too difficult or too competitive, and found their CS programs comparatively much easier mathematically, with most CS fields requiring much less background to get to the academic cutting edge (as should be expected for a much younger and more application focused field), with an easier path for newcomers to publish meaningful results in high-impact journals. YMMV.
Ok, I should quantify it as "some of world's Top 10 engineering universities" then. YMMV as you say; imagine you are required to read 100 papers a term in a single subject in CS these days. Moreover, you should be almost assured that most of the facts you learn will be/are already obsolete due to crazy pace CS is having in some areas. I am not aware of math/physics having such a crazy pace, but I might be ignorant there.
Except for the “borderline mathematics” parts of computer science (algorithms, PL semantics, etc.), the scientific rigor in CS publications is much, much, much lower than in math or physics publications. Scientific progress is to be measured in terms of advances in human understanding, not in number of publications, our current “publish or perish” culture notwithstanding.
Again, if you want to coast, collect grade inflated Bs and Cs, you can do without much rigor. If you want As and are ambitious, you have to master both CS and related math. Publish or perish is terrible, I agree; if you focus only on super hard high risk problems that might advance humanity, you'll get kicked out of university in no time. And as a professor, 90% of your time will be spent on chasing grants.
I am talking about scientific rigor, not grades. A more relevant question (at least to my interests) is “What tests does a scientific proposal have to pass to become an estabilshed scientific result?”
