In order to apply Propositions 13.1, 13.2, and 13.3, we need a useful criterion that tells us when a series of functions converges uniformly over a set S. Weierstrass gave such a criterion for …

The functions fi are all continuous functions on a compact set (hence uniformly continuous by Theorem 1.2). If fi ! f uniformly, then f is continuous, however we know that the limit f is not …

This example shows that a function can be uniformly contin-uous on a set even though it does not satisfy a Lipschitz inequality on that set, i.e. the method of Theorem 8 is not the only method …

The same as true for each fn by Cauchy's integral formula, which together with the proof of (1) above establishes that ff0 n(z)g1 n=0 converges uniformly to f0(z).

Note: The above proof also shows that if fn are uniformly continuous and fn → f uniformly, then f is uniformly continuous. The condition |x − y| < 1/k in the proof can be removed and all …

Intuitively, gn(x) → g(x) uniformly if it is possible to draw an -band around the graph of g(x) that contains all of the graphs of gn(x) for large enough n.

Theorem. The series P1 n=1 un(z) is uniformly convergent on a compact subset only if the following holds. For each > 0 there is a value n0 depending on R such that the p rtial sums Sm z)