Uniform convergence

A sequence of functions $f_n:X\to Y$ converges uniformly when one index $N$ makes $f_n(x)$ close to the limit simultaneously for every $x\in X$.

If continuously differentiable functions have uniformly convergent derivatives on an interval and their values converge at one point, the functions converge locally uniformly to a continuously differentiable limit whose derivative is the derivative limit. The proof integrates from the anchored point using the fundamental theorem of calculus. Uniform convergence of the functions on the entire interval follows when the interval is bounded; it need not follow on an unbounded interval.

On a finite-measure measurable set, a sequence of measurable functions converging pointwise almost everywhere converges uniformly outside a set of arbitrarily small measure. For tolerance $1/k$, the sets on which every index after $N$ has error at most $1/k$ increase to full measure as $N\to \infty$. Choose $N_k$ with exceptional measure at most $\epsilon 2^{-k-1}$ and intersect the good sets. Countable subadditivity bounds the discarded measure, and the resulting tail estimates prove uniform convergence. Finite measure is essential: $1_{[n,\infty )}$ on $[0,\infty )$ is a counterexample.

A uniform limit of continuous functions is continuous.

Dini's theorem says that a monotone sequence of continuous real-valued functions on a compact space that converges pointwise to a continuous function converges uniformly.

A sequence of functions is uniformly Cauchy when, for every $\epsilon >0$, all sufficiently late pairs satisfy

A sequence $f_n:X\to Y$ converges locally uniformly to $f$ when every point has a neighbourhood on which the convergence is uniform.

The compact-open topology on a space of functions is generated by uniform control on compact subsets. For scalar-valued functions, its basic seminorms are $f\mapsto {\sup }_{x\in K}|f(x)|$ as $K$ ranges over compact subsets of the domain.

A locally uniform limit of continuous real-valued functions is continuous. Moreover, local uniform convergence is uniform on every compact subset: choose finitely many neighbourhoods from the local uniformity cover and take the largest of their convergence indices.