Classes of functions

We first met a sequence as a particularly easy sort of function, defined on $\mathbb {N}$, rather than $\mathbb {R}$. We now move to the more general study of functions. However, our earlier work wasn't a diversion; we shall see that sequences prove a useful tool both to investigate functions, and to give an idea of appropriate methods.

Our main reason for being interested in studying functions is as a model of some behaviour in the real world. Typically a function describes the behaviour of an object over time, or space. In selecting a suitable class of functions to study, we need to balance generality, which has chaotic behaviour, with good behaviour which occurs rarely. If a function has lots of good properties, because there are strong restrictions on it, then it can often be quite hard to show that a given example of such a function has the required properties. Conversely, if it is easy to show that the function belongs to a particular class, it may be because the properties of that class are so weak that belonging may have essentially no useful consequences. We summarise this in the table:

Strong restrictions Weak restrictions

Good behaviour Bad behaviour

Few examples Many examples

It turns out that there are a number of ``good'' classes of functions which are worth studying. In this chapter and the next ones, we study functions which have steadily more and more restrictions on them. Which means the behaviour steadily improves; and at the same time, the number of examples steadily decreases. A perfectly general function has essentially nothing useful that can be said about it; so we start by studying continuous functions, the first class that gives us much theory.

In order to discuss functions sensibly, we often insist that we can ``get a good look'' at the behaviour of the function at a given point, so typically we restrict the domain of the function to be well behaved.

Definition 4.1   A subset U of $\mathbb {R}$ is open if given a $\in$ U, there is some $\delta$ > 0 such that (a - $\delta$, a + $\delta$) $\subseteq$ U.

In fact this is the same as saying that given a $\in$ U, there is some open interval containing a which lies in U. In other words, a set is open if it contains a neighbourhood of each of its points. We saw in 1.10 that an open interval is an open set. This definition has the effect that if a function is defined on an open set we can look at its behaviour near the point a of interest, from both sides.