Which mathematical theory is primarily concerned with the study of functions involving real numbers?

Real Analysis is fundamentally focused on the study of functions involving real numbers. It delves into the properties, limits, continuity, and integrability of functions defined on real number domains. This field rigorously explores concepts such as sequences, series, and the distinction between different types of convergence, enabling a deep understanding of the behavior of real-valued functions.

In comparison, Function Theory is a broader term that may encompass various types of functions, not strictly limited to those involving real numbers, and doesn't specifically identify with studying real numbers alone. Complex Analysis focuses on functions of complex numbers, exploring properties unique to the complex plane which are not applicable in the realm of purely real numbers. Linear Algebra deals with vector spaces and linear mappings between these spaces, primarily focusing on matrices and their properties rather than the intricacies of real-valued functions.

Thus, Real Analysis stands out as the most relevant theory regarding functions specifically involving real numbers, making it the correct answer.