Uniform convergence

Mode of convergence of a function sequence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n) {\displaystyle (f_{n})} converges uniformly to a limiting function f {\displaystyle f} if, roughly speaking, they uniformly approximate the function f {\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 f {\displaystyle f} , the graphs of all but finitely many of the functions f n {\displaystyle f_{n}} lie within that thin band.

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