What theory focuses on limits, continuity, and infinitesimal calculus?

Real Analysis is the branch of mathematics that rigorously investigates the properties and behavior of real numbers, sequences, and functions. It specifically deals with concepts such as limits and continuity, which are fundamental to understanding calculus. In Real Analysis, limits are used to define continuity, derivatives, and integrals, providing a solid foundation for infinitesimal calculus.

The theory also explores convergence and divergence of sequences and series, which are essential to understanding how functions behave over different intervals and how they can be represented through infinite sums. Furthermore, it delves into the properties of functions and discusses various aspects of metric spaces, which enhance the understanding of continuity in a more abstract setting.

While other options such as Probability Theory, Function Theory, and Complex Analysis have their own important focuses, they do not center primarily on the foundational concepts of limits and continuity as extensively as Real Analysis does. Probability Theory focuses on the mathematical framework for dealing with randomness and uncertainty, Function Theory typically deals with properties of functions, often holomorphic functions in some contexts, and Complex Analysis focuses on functions of complex variables.