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 ${\displaystyle f}$ on a set ${\displaystyle E}$ as the function domain if, given any arbitrarily small positive number ${\displaystyle \varepsilon }$, a number ${\displaystyle N}$ can be found such that each of the functions ${\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }$ differs from ${\displaystyle f}$ by no more than ${\displaystyle \varepsilon }$ at every point ${\displaystyle x}$ in ${\displaystyle E}$. That is, the required value of ${\displaystyle N}$ only depends on ${\displaystyle \varepsilon ,}$ not on any particular ${\displaystyle x}$.

By contrast, pointwise convergence of ${\displaystyle (f_{n})}$ to ${\displaystyle f}$ merely guarantees that given any ${\displaystyle x\in E}$, we can find ${\displaystyle N=N(\varepsilon ,x)}$ (that is, possibly depending on ${\displaystyle x}$ as well as ${\displaystyle \varepsilon }$) such that for the specific value of ${\displaystyle x}$ given, ${\displaystyle f_{n}(x)}$ falls within ${\displaystyle \varepsilon }$ of ${\displaystyle f(x)}$ whenever ${\displaystyle n\geq N.}$ A different ${\displaystyle x}$ may require a larger value of ${\displaystyle N.}$

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 ${\displaystyle f_{n}}$, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit ${\displaystyle f}$ if the convergence is uniform, but not necessarily if the convergence is not uniform.