Uniform convergence
Undergraduate
Consider a function $f$ and a sequence of functions $f_n$ on a set $X$. The sequence $f_n$ converges uniformly to $f$ if, for every $\varepsilon > 0$, there exists $N \in \nats$ such that
\[\abs{f_n(x) - f(x)} < \varepsilon \quad \text{for all } n \geq N \text{ and all } x \in X.\]Properties
- If the functions $f_n$ are continuous and converge uniformly to $f$, then $f$ is continuous. If $f_n$ converges pointwise to $f$ instead of uniformly, then $f$ need not be continuous: $f_n(x) = x^n$ on $[0, 1]$ converges pointwise to the function that is $0$ for $x < 1$ and $1$ at $x = 1$.
Notes
Pointwise convergence only requires $f_n(x) \to f(x)$ at each $x$ separately, so $N$ may depend on $x$.