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 p_n ∈ A such that |f(p_n)| >n. Since A is compact, the sequence {p_n}⊂ A must have a convergent subsequence, say {q_n}, and q_n → p¯  with  p¯ ∈ A. Since f is Continuous on A, we have f(q_n) → f(p¯) as n →∞. Choosing ε = 1in the definition of continuity on A and observing that d(q_n, p¯) → 0as n →∞, we can state that there is an N1 such that for n>N₁, we have

                                                |f(q_n) − f(p ¯)| < 1    whenever  n>N₁.

We choose N₁ so large that |f(p ¯)| <N₁. Now for n>N₁ we may write

Since q_n is at least the nth member of the sequence {p_n}, it follows that |f(q_n)| >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.

1. If c is a number and f(p)= c for all p ∈ A, then show that for any limit point p₀ of A, we have

           (theorem on limit of a constant).

      Basic Elements of Real Analysis, Murray H. Protter, Springer, 1998 .Page(122-124)