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Typically in the US, the calculus sequence is one semester differentiation, one semester integration, and a third semester of three dimensional and vector calculus. The × symbol is used a lot for vector cross products in the third semester. Typically these courses don't involve proofs. Serious students frequently take a portion of this sequence +/- matrix algebra in high school as AP courses or dual enrollment where the school cooperates with a local college to share their exams and get official credit. They are technically considered to be college level courses in the US. I think a lot of the content in them is covered in A level further maths or IB HL math or whatever your local equivalent is.

This sequence is followed by differential equations courses for the physicists, engineers, and most mathematics majors. Then every college has a mechanism to generate mathematical maturity in their first or second year pure math majors - sometimes it's a proof focused version of linear algebra, sometimes it's a specific Introduction to Proofs course, sometimes it's a discrete math/set theory course, sometimes it's groups/rings or real analysis but slowed down a bit at first. This gates the upper level pure mathematics courses, where most programs require one semester each of algebra and analysis and some number of elective courses.

A general definition of continuity typically doesn't arise until a topology course or a second semester real analysis course. It is entirely possible to graduate from most mathematics bachelor's programs in the US without taking either of those courses.