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Date: 23-11-2016
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In Chapter of(Basic Properties of Functions on R1), we discussed the elementary properties of functions defined on R1 with range in R1. We were also concerned with functions whose domain consisted of a part of R1, usually an interval. In this section we develop properties of functions with domain all or part of an arbitrary metric space S and with range in R1. That is, we consider real valued functions. We first take up the fundamental notions of limits and continuity.
Definition
Suppose that f is a function with domain A, a subset of a metric space S, and with range in R1; we write f : A → R1. We say that f(p) tends to J asP tends toP0 through points of A if (i) p0 is a limit point of A, and (ii) for each ε> 0 there is a δ> 0 such that
|f(p) − l| <ε for all p in A
with the property that 0 <d(p,p0)<δ. We write
f(p) → l as p → p0,p ∈ A.
We shall also use the notation
Observe that the above definition does not require f(p0) to be defined, nor does p0 have to belong to the set A. However, when both of these conditions hold, we are able to define continuity for real-valued functions.
Definitions
Let A be a subset of a metric space S, and suppose f : A → R1 is given. Let p0 ∈ A. We say that f is continuous with respect to A at p0 if (i) f(p0) is defined, and (ii) either p0 is an isolated point of A or p0 is a limit point of A and
f(p) → f(p0) as p → p0,p ∈ A.
We say that f is continuous on A if f is continuous with respect to A at every point of A. If the domain of f is the entire metric space S, then we say that f is continuous at p0, omitting the phrase “with respect to A.” The definitions of limit and continuity for functions with domain in one metric space S1 and range in another metric space S2 are extensions of those given above. The basic properties of functions defined on R1 discussed do not always have direct analogues for functions defined on a subset of a metric space. For example, the Intermediate-value theorem ( (Intermediate-value theorem) Suppose f is continuous on [a,b], c ∈ R1, f(a)<c, and f(b)>c. Then there is at least one number x0 on [a,b] such that f(x0) = c.)) for functions defined on an interval of R1 does not carry over to real-valued functions defined on all or part of a metric space. However, when the domain of a real-valued function is a compact subset of a metric space, For example, the following theorem is the analogue of the boundedness theorem ( (Boundedness Theorem) Suppose that the domain of f is the closed interval I ={x:a ≤ x ≤ b}, and f is continuous on I. Then the range of f is bounded.).
Theorem 1.1
Let A be a compact subset of a metric space S. Suppose that f :A → R1 is continuous on A. Then the range of f is bounded.
Proof
We assume that the range is unbounded and reach a contradiction. Suppose that for each positive integer n, there is a pn ∈ A such that |f(pn)| >n. Since A is compact, the sequence {pn}⊂ A must have a convergent subsequence, say {qn}, and qn → p¯ with p¯ ∈ A. Since f is Continuous on A, we have f(qn) → f(p¯) as n →∞. Choosing ε = 1in the definition of continuity on A and observing that d(qn, p¯) → 0as n →∞, we can state that there is an N1 such that for n>N1, we have
|f(qn) − f(p ¯)| < 1 whenever n>N1.
We choose N1 so large that |f(p ¯)| <N1. Now for n>N1 we may write
Since qn is at least the nth member of the sequence {pn}, it follows that |f(qn)| >n for each n. Therefore,
a contradiction for n sufficiently large. Hence the range of f is bounded.
Note the similarity of the above proof to that of (Boundedness Theorem). Also, observe the essentialmanner in which the compactness of A isemployed. The result clearly does not hold if A is not compact.
Problems
In the following problems a set A in ametric space S is given. All functions are real-valued (range in R1) and have domain A.
(theorem on limit of a constant).
Basic Elements of Real Analysis, Murray H. Protter, Springer, 1998 .Page(122-124)
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