Analysis is mathematics done with limits — the rigorous study of the infinitely small and the infinitely close. It is where calculus grows up: the intuitive ideas of rate, slope and area are put on unshakeable foundations, and then pushed to places intuition can no longer follow. Nearly all of the mathematics that describes the physical world lives here.

The areas to explore

Start with calculus — limits, derivatives, integrals, series, and the differential equations that model change (and, in its analysis chapters, the ε–δ rigour underneath it all). Cross into the imaginary with complex analysis, where demanding a function be differentiable just once makes it perfect forever. Rebuild the idea of "area" from scratch in measure theory, the Lebesgue integral and the true foundation of probability. And take calculus into infinite dimensions with functional analysis, where whole functions become the "points" of a space.