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 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 the rate at which approaches is "uniform" throughout its domain in the following sense: in order to guarantee that falls within a certain distance of , we do not need to know the value of in question — there can be found a single value of independent of , such that choosing will ensure that is within of for all . In contrast, pointwise convergence of t