Real Functions: Continuity Theorem

This lecture covers the Continuity Theorem for functions dependent on a parameter, proving the continuity of a function g defined as g(x) = f(t, x) for a given function f. The demonstration involves showing the uniform continuity of f within a certain interval, leading to the conclusion that g is continuous. The proof is detailed step by step, emphasizing the conditions under which g remains continuous. Through a series of mathematical manipulations and constraints, the instructor illustrates the concept of continuity in real functions.