## Definition 8.2.1: Uniform Convergence

*{ f*with domain

_{n}(x) }*converges uniformly to a function*

**D***f(x)*if given any

*> 0*there is a positive integer

*N*such that

| f_{n}(x) - f(x) | < for allxwheneverDn N

Please note that the above inequality must hold *for all x
in the domain*, and that the integer

*N*depends only on .

We should compare uniform with pointwise convergence:

- For
pointwiseconvergence we couldfirstfix a value forxandthenchooseN. Consequently,Ndepends on both andx.- For
uniformconvergencefmust be_{n}(x)uniformlyclose tof(x)forallxin the domain. ThusNonly depends on but not onx.

Let's illustrate the difference between pointwise and uniform convergence graphically:

Pointwise Convergence Uniform Convergence For pointwise convergence we first fix a value

x. Then we choose an arbitrary neighborhood around_{0}f(x, which corresponds to a vertical interval centered at_{0})f(x._{0})

Finally we pick

Nso thatfintersects the vertical line_{n}(x_{0})x = xinside the interval_{0}(f(x_{0}) - , f(x_{0}) + )For uniform convergence we draw an -neighborhood around the

entirelimit functionf, which results in an "-strip" withf(x)in the middle.

Now we pick

Nso thatfis completely inside that strip_{n}(x)for all.xin the domain