It's just that I can't quite imagine the IB FM Calculus course (and it wasn't at my old school) being anywhere near as thorough a proper analysis courses where you construct the rationals and then construct the reals and talk about convergence and what not whereas the IB course (in the way that I was taught it, at least) was more 'this is true and this is true and we're going to assume the reals exist btw forget about dedekind cuts m8'. I will have to admit that I'm slightly biased against the IB system given that I was in it for a year and thoroughly disliked my experience. It's not unpleasant at all, I very much appreciate you sharing your opinion. Like you said, the IB FM Calculus option covers a heck of the analysis first year course, but all the rigour is missing, which is a fundamental part of analysis.

I would argue that CIE is the better route as it sets you up better for getting into uni (doing well in STEP) and you'll be able to learn advanced maths there than learning a bunch of advanced maths in IB FM HL which is taught horribly, missing out on STEP and then if they manage to scrape into uni, having to unlearn the bad unrigorous habits taught at IB and relearn them properly and rigorously at uni. But none of the stuff the CIE misses out on but is taught in IB is part of the STEP syllabus. Things like polar coordinates, parametric equations, and a bunch of stuff that I've forgotten. Plus, IB misses out on a fair bit of content that is covered by CIE and is used in the STEP examinations. You are taught advanced university level material in a formulaic and secondary-school natured way that you end up having to unlearn at uni and re-learn properly. I think the IB Futher maths is horribly taught. If I really had to choose the 'harder' route then I would say the IB double Math route is possibly the more challenging. I teach them both and would say that either route would set you up in good stead for a UK based degree in Math. The bottom line is that the approaches to Maths are very different, both cover a good whack of content, but there is a different emphasis. (all manner of sequences, series, convergence tests, proofs associated with the aforementioned MVT, L'hopital, Rolle, Taylor including a proper analysis of error functions etc. Just considering one module, eg FM Calculus option is very analytical in nature and covers topics met often in a 1st year undergrad Analysis course.

The HL maths + HL Further Maths combination covers one heck of a lot of material.

Further Maths Ib Textbook