The epsilon-delta definition is a formalization of the limit concept in calculus, providing a rigorous foundation for the notion of continuity and convergence. It asserts that a function f(x) approaches a limit L as x approaches a point c if, for every positive number ε, there exists a positive number δ such that whenever 0 < |x - c| < δ, it follows that |f(x) - L| < ε.