Uniform convergence is a type of convergence for sequences of functions. A sequence of functions f_n converges uniformly to a function f on a set D if, for every small positive number ε, there exists a natural number N such that for all n ≥ N and for all x in D, the difference |f_n(x) - f(x)| is less than ε. This means that the functions get uniformly close to f across the entire set D. This concept is important in analysis because it ensures that certain properties, like continuity and integration, are preserved in the limit. If a sequence of continuous functions converges uniformly to a function, then that limit function is also continuous. This is a stronger condition than pointwise convergence, where the convergence can vary at different points in the domain.