Let XXX be a set and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space. A sequence of functions fn:X→Yf_n:X\to Yfn​:X→Y is uniformly Cauchy if ∀ε>0, ∃N∈N such that ∀m,n≥N, ∀x∈X, dY(fm(x),fn(x))<ε.\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall m,n\ge N,\ \forall x\in X,\ d_Y\bigl(f_m(x),f_n(x)\bigr)<\varepsilon.∀ε>0, ∃N∈N such that ∀m,n≥N, ∀x∈X, dY​(fm​(x),fn​(x))<ε. Equivalently, ...

Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces , let E⊆XE\subseteq XE⊆X, and let f:E→Yf:E\to Yf:E→Y. The function fff is uniformly continuous on EEE if ∀ε>0, ∃δ>0 such that ∀x,y∈E, (dX(x,y)<δ⇒dY(f(x),f(y))<ε).\forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall x,y\in E,\ \bigl(d_X(x,y)<\delta \Rightarrow d_Y(f(x),f(y))<\varepsilon\bigr).∀ε>0, ∃δ>0 such that ∀x,y∈E, (dX​(x,y)<δ⇒dY​(f(x),f(y))<ε).Uniform continuity strengthens pointwise continuity by requiring that the same δ\deltaδ works everywhere on EEE. It is essential for interchanging limits and integrals and for extension theorems; continuous functions on compact sets are uniformly continuous. ...

Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces and let f:X→Yf:X\to Yf:X→Y. Proposition: If fff is uniformly continuous on XXX, then fff is continuous at every point of XXX. Uniform continuity is strictly stronger than continuity in general, but on compact sets they coincide (Heine–Cantor ). ...

Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces and let f:X→Yf:X\to Yf:X→Y be uniformly continuous . Proposition: If (xn)(x_n)(xn​) is a Cauchy sequence in XXX, then (f(xn))(f(x_n))(f(xn​)) is a Cauchy sequence in YYY. ...

Let XXX be a set and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space . A sequence of functions fn:X→Yf_n:X\to Yfn​:X→Y converges uniformly to f:X→Yf:X\to Yf:X→Y if ∀ε>0, ∃N∈N such that ∀n≥N, ∀x∈X, dY(fn(x),f(x))<ε.\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n\ge N,\ \forall x\in X,\ d_Y\bigl(f_n(x),f(x)\bigr)<\varepsilon.∀ε>0, ∃N∈N such that ∀n≥N, ∀x∈X, dY​(fn​(x),f(x))<ε. Equivalently, ...

Let XXX be a set and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space. A series of functions ∑n=1∞fn\sum_{n=1}^\infty f_n∑n=1∞​fn​ converges uniformly on XXX if the sequence of partial sums SN(x):=∑n=1Nfn(x)S_N(x):=\sum_{n=1}^N f_n(x)SN​(x):=n=1∑N​fn​(x) converges uniformly to a limit function S:X→YS:X\to YS:X→Y. ...

Uniform convergence and differentiation: Let fn:[a,b]→Rf_n:[a,b]\to\mathbb{R}fn​:[a,b]→R be differentiable on [a,b][a,b][a,b] (or on (a,b)(a,b)(a,b) with suitable endpoint control). Assume: there exists x0∈[a,b]x_0\in[a,b]x0​∈[a,b] such that the sequence (fn(x0))(f_n(x_0))(fn​(x0​)) converges in R\mathbb{R}R, and the derivatives fn′f_n'fn′​ converge uniformly on [a,b][a,b][a,b] to a function ggg. Then fnf_nfn​ converges uniformly on [a,b][a,b][a,b] to a differentiable function fff, and f′(x)=g(x)for all x∈[a,b]. f'(x)=g(x)\quad \text{for all } x\in[a,b]. f′(x)=g(x)for all x∈[a,b]. ...

Uniform convergence and integration (Riemann): Let fn:[a,b]→Rf_n:[a,b]\to\mathbb{R}fn​:[a,b]→R be Riemann integrable for each nnn, and suppose fn→ff_n\to ffn​→f uniformly on [a,b][a,b][a,b]. Then: fff is Riemann integrable on [a,b][a,b][a,b], and the integrals converge to the integral of the limit: lim⁡n→∞∫abfn(x) dx=∫abf(x) dx. \lim_{n\to\infty}\int_a^b f_n(x)\,dx=\int_a^b f(x)\,dx. limn→∞​∫ab​fn​(x)dx=∫ab​f(x)dx. This theorem justifies passing limits through integrals when convergence is uniform, and it is a standard tool in approximation and series-of-functions arguments. ...

Let XXX be a set and let fn,f:X→Rf_n,f:X\to\mathbb{R}fn​,f:X→R (or C\mathbb{C}C). Define the sup norm (when finite) by ∥h∥∞=sup⁡x∈X∣h(x)∣. \|h\|_\infty=\sup_{x\in X} |h(x)|. ∥h∥∞​=supx∈X​∣h(x)∣. Uniform convergence and sup norm: The sequence fnf_nfn​ converges uniformly to fff on XXX if and only if ∥fn−f∥∞→0. \|f_n-f\|_\infty \to 0. ∥fn​−f∥∞​→0. Moreover, whenever the sup norms are finite, ∣∥fn∥∞−∥f∥∞∣≤∥fn−f∥∞. \bigl|\|f_n\|_\infty-\|f\|_\infty\bigr|\le \|f_n-f\|_\infty. ​∥fn​∥∞​−∥f∥∞​​≤∥fn​−f∥∞​. ...

Corollary: Let XXX be a set and let (fn)(f_n)(fn​) be functions fn:X→Yf_n:X\to Yfn​:X→Y into a metric space (Y,d)(Y,d)(Y,d). If fn→ff_n\to ffn​→f uniformly on XXX, then fn→ff_n\to ffn​→f pointwise on XXX; i.e., for every x∈Xx\in Xx∈X, fn(x)→f(x). f_n(x)\to f(x). fn​(x)→f(x). ...