Funksjonskonvergens

Funksjonskonvergens is a concept in mathematics that describes how a sequence of functions can approach a limit function. This means that as the index of the sequence increases, the individual functions in the sequence become increasingly similar to a specific function, known as the limit function. There are two primary types of convergence for functions: pointwise convergence and uniform convergence.

Pointwise convergence is the simpler of the two. A sequence of functions fn(x) converges pointwise to a function f(x) on a set S if for every point x in S, the sequence of real numbers fn(x) converges to the real number f(x). This means that for each individual x, the values of the functions get arbitrarily close to the value of the limit function at that point.

Uniform convergence is a stronger condition. A sequence of functions fn(x) converges uniformly to a function f(x) on a set S if the rate at which fn(x) approaches f(x) is independent of x. More formally, for any epsilon greater than zero, there exists an N such that for all n greater than N, the absolute difference between fn(x) and f(x) is less than epsilon for all x in S. This implies that the "gap" between the functions in the sequence and the limit function is uniformly small across the entire set S. Uniform convergence has important implications for the continuity and integrability of the limit function, often preserving these properties from the sequence functions.