In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in . Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance , we only need to make sure that is larger than or equal to a certain , which we can find without knowing the value of in advance. In other words, there exists a number that could depend on but is independent of , such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find (i.e., could depend on the values of both and ) such that, for that particular , falls within of whenever (and a different may require a different, larger for to guarantee that ).
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions , such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit if the convergence is uniform, but not necessarily if the convergence is not uniform.