Uniform convergence
A sequence of functions 
is said to uniformly converge to a function 
if for every positive real number 
, then there exists 
such that for all positive integers 
, we have 
. (More generally, we can replace 
with any metric space 
.)
This is different from pointwise convergence, where a sequence of functions converge pointwise if at every point in the domain, the functions converge. Uniform convergence is a stronger condition, because (speaking informally) the function has to converge at a similar rate everywhere on its domain.
![[asy] size(150); import graph; real epsilon = 0.1; pen d = linewidth(0.7); defaultpen(d); // implement [an apparently non-functional] version of Lagrange. real symproduct(real a, real x[]){ real product = 1; for(int i = 0; i < x.length; ++i) if(a != x[i]) product = product * (a - x[i]); return product; } real baryweight(int i, real x[]){ real product = 1; for(int j = 0; j < x.length; ++j) if(j != i) product = product * (x[j] - x[i]); return 1/product; } real lagrange(real a, real x[], real y[]){ real sum = 0; for(int i = 0; i < x.length; ++i) if(a != x[i]) sum = sum + baryweight(i,x)*y[i]/(a - x[i]); return symproduct(a, x) * sum; } real func(real x){ // note there is a sign error. use odd # of points real LISTx[] = {-1.5, -0.5 , 0 ,0.4, 0.8, 1, 4}; real LISTy[] = {-4, -0.5 , 0.2 , 0.5, 0.35, 0.6, 1.2}; return lagrange(x,LISTx,LISTy); } real func_plus(real x) { return (func(x) + epsilon); } real func_minus(real x) { return (func(x) - epsilon); } draw(graph(func,-0.00001,1.00001),linewidth(1.2)); draw(graph(func_plus,-0.00001,1.00001),dashed+linewidth(0.7)); draw(graph(func_minus,-0.00001,1.00001),dashed+linewidth(0.7)); // convergents real func_conv1(real x) { return (func(x) + 1.5*epsilon*sin(12*x+pi/3)); } real func_conv2(real x) { return (func(x) + 0.9*epsilon*sin(12*x+pi/4)); } real func_conv3(real x) { return (func(x) + 0.5*epsilon*sin(12*x+pi/5)); } draw(graph(func_conv1,-0.00001,1.00001),red+d); draw(graph(func_conv2,-0.00001,1.00001),green+d); draw(graph(func_conv3,-0.00001,1.00001),blue+d); // draw axes Label f; f.p=fontsize(8); xaxis(0, 1, Ticks(f, 1, 1)); yaxis(0, 1, Ticks(f, 1, 1)); pair a = (-0.1,func(0.00001)); draw(a--a+(0,epsilon), Arrows(6)); draw(a--a-(0,epsilon), Arrows(6)); label("$\varepsilon$",a+(0,epsilon/2),W,f.p); label("$\varepsilon$",a-(0,epsilon/2),W,f.p); [/asy]](http://latex.artofproblemsolving.com/9/f/e/9fe11cc837896d059fe3d29973350f11cc16864e.png)
If the functions
continue to approach 
as shown, they converge uniformly.Every uniformly convergent sequence converges pointwise, but the converse is not necessarily true. For example, the sequence of functions defined by 
for ![$x \in [0, 1]$](http://latex.artofproblemsolving.com/a/f/e/afe25008185bb22130658a17e0903591635b60ed.png)
converges pointwise to the function 
, but this convergence is not uniform.
Properties
An equivalent definition is that if 
for all 
and
![\[\lim_{n \to \infty} \sup_{x \in X} |f(x) - f_n(x)| = 0,\]](http://latex.artofproblemsolving.com/3/7/7/377a2a1b15f10625222281ce11b0cb6db5947d9e.png)
then 
converges uniformly.
Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have. A uniformly convergent sequence of continuous functions converges to a continuous function. A uniformly convergent sequence of differentiable functions defined on a closed interval converges to a differentiable function, and a sequence of Stieltjes-integrable functions converges to a Stieltjes-integrable function. It is possible to show by example that these properties do not have to hold for pointwise convergent functions.
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