My father (and me) would always recommend Zeldovich's "Higher mathematics for beginners" for learning analysis at the upper high-school level. This particular book does not seem to be available in translation, instead there is a reworked version with Yaglom (who was a brilliant science educator himself):

https://archive.org/details/MIRZeldovichYAndYaglomIHigherMat...

Zeldovich's book with Myshkis on applied mathematics is also excellent: https://archive.org/details/ZeldovichMyskisElementsOfApplied...

Subjectively, I prefer typesetting of the latter, but that is because it is closer to the original Russian edition. Zeldovich was a physicist, so these take an engineer's/physicist's approach, which is, in my opinion, the right entry point to analysis. The reader effectively has to follow the historic development of the subject, starting with some intuitive observations, and eventually developing quite delicate insights.

I read HMFB when I was about 17, and it was great. I remember making up questions of the sort "What level of soda in a can makes it the most stable", and the like, inspired by the book.