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and to give an idea of appropriate methods. Our main reason for being interested in studying functions is as a model of some beha- viour 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. 4.1. Definition. A subset U of R is open if given a " U, there is some ´ >0 such that (a - ´, a + ´) †" U. In fact this is the same as saying that given a " 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 29 30 CHAPTER 4. LIMITS AND CONTINUITY 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. 4.2 Limits and Continuity We discuss a number of functions, each of which is worse behaved than the previous one. Our aim is to isolate an imprtant property of a function called continuity. 4.2. Example. 1. Let f(x) =sin(x). This is defined for all x " R. [Recall we use radians automatically in order to have the derivative of sin x being cos x.] 2. Let f(x) =log(x). This is defined for x>0, and so naturally has a restricted domain. Note also that the domain is an open set. x2 -a2 3. Let f(x) = when x = a, and suppose f(a) =2a. x-a sin x when x =0, 4. Let f(x) = x =1 if x=0. 5. Let f(x) =0 if x 1 6. Let f(x) =sin when x =0 andlet f(0) = 0. x In each case we are trying to study the behaviour of the function near a particular point. In example 1, the function is well behaved everywhere, there are no problems, and so there is no need to pick out particular points for special care. In example 2, the function is still well behaved wherever it is defined, but we had to restrict the domain, as promised in Sect. 1.5. In all of what follows, we will assume the domain of all of our functions is suitably restricted. We won t spend time in this course discussing standard functions. It is assumed that you know about functions such as sin x, cos x, tan x, log x, exp x, tan-1 x and sin-1 x, as well as the obvious ones like polynomials and rational functions those functions of the form p(x)/q(x), where p and q are polynomials. In particular, it is assumed that you know these are differentiable everywhere they are defined. We shall see later that this is quite a strong piece of information. In particular, it means they are examples of continuous functions. Note also that even a function like f(x) =1/x is continuous, because, wherever it is defined (ie on R \{0}), it is continuous. In example 3, the function is not defined at a, but rewriting the function x2 - a2 = x + a if x = a, x - a we see that as x approaches a, where the function is not defined, the value of the function approaches 2a. It thus seems very reasonable to extend the definition of f by defining f(a) =2a. In fact, what we have observed is that x2 - a2 [ Pobierz caÅ‚ość w formacie PDF ] |
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