The most basic object of study in mathematics is of a process. Processes take in input and spew out output. Almost everything that one wants to study scientifically can be modelled mathematically by a process. Given a process, there are three types of question that one wants to answer:

1. What I put in X, what did I get out?
2. Where I got out Y, where did I start?
3. How I put in X, I got out Y, how did I get it?

We model processes in mathematics by functions. Thus the study of processes leads one, in mathematics, to the study of functions. In the first two questions, the function is (presumably) known and the questions are about studying it. The last question in the list above is concerned with the question of finding a function. To do this, one must have some idea of the types of function that might fit, how these behave, and how to describe them. This involves studying not just one function but whole families of functions.

One of the simplest such cases is the family of continuous functions on the interval. One can think of these as modelling processes whereby:

1. The input is a parameter between 0 and 1.
2. The output is a (real or complex) number.
3. The output depends continuously on the input.

In this course, we shall develop the tools necessary to study such functions and, more importantly, the space of such functions.