Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ${\displaystyle (f_{n})}$ converges uniformly to a limiting function ${\displaystyle f}$ if, roughly speaking, they uniformly approximate the function ${\displaystyle f}$ over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar of the original function. Graphically this means that, given any thin band around the graph of ${\displaystyle f}$, the graphs of all but finitely many of the functions ${\displaystyle f_{n}}$ lie within that thin band. This is in contrast to pointwise convergence, in which all but finitely many of the functions lie in a thin band at each point, but the finite set of functions which must be excluded in order for that to be the case varies from point to point.

The strength of uniform convergence makes it ideal in many applications, where pointwise convergence is not sufficient. For example, the uniform limit of a sequence of continuous functions is automatically continuous; the uniform limit of Riemann integrable functions is automatically Riemann integrable. With additional hypotheses, differentiability can be transferred to the limit function as well. 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 was first formalized by Karl Weierstrass.