We know what it means to say that a sequence or series of numbers converges. In many applications we will deal with sequences and series of functions. For instance, you have already seen Taylor Series in Calculus, and may have seen Fourier Series in a course on Pde's. We first define just what we mean by convergence in the case of functions.
Uniform continuity is a useful way to characterize certain functions.
We saw that when a function is continuous on a compact set then it is uniformly continuous there. Similarly, we will now define a type of convergence for sequences of functions, called uniform convergence, which will prove to be useful when we wish to preserve properties in the limit. We will need to measure the distance between two functions. There are various ways to do this, many of which involve integrals. One simple way derives from the following.