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2 Limit, Continuity and Differentiability of Functions 2 Limit, Continuity and Differentiability of Functions In this chapter we shall study limit and continuity of real valued functions defined on certain sets. 2.1 Limit of a Function Suppose f is a real valued function defined on a subset D of R. We are going to define limit of f(x) as x 2 D approaches a point a which is not necessarily in D. First we have to be clear about what we mean by the statement \x 2 D ap- proaches a point a". 2.1.1 Limit point of a set D ⊆ R Definition 2.1 Let D ⊆ R and a 2 R. Then a is said to be a limit point of D if for any δ > 0, the interval (a − δ; a + δ) contains atleast one point from D other than possibly a, i.e., D \ fx 2 R : 0 < jx − aj < δg 6= ?: Example 2.1 The statements in the following can be easily verified: (i) Every point in an interval is its limit point. (ii) If I is an open interval of finite length, then both the end points of I are limit points of I. (iii) The set of all limit points of an interval I of finite length consists of points from I together with its endpoints. (iv) If D = fx 2 R : 0 < jxj < 1g, then every point in the interval [−1; 1] is a limit point of D. (v) If D = (0; 1) [ f2g, then 2 is not a limit point of D. The set of all limit points of D is the closed interval [0; 1]. 43 44 Limit, Continuity and Differentiability of Functions M.T. Nair 1 (vi) If D = f n : n 2 Ng, then 0 is the only limit point of D. (vii) If D = fn=(n + 1) : n 2 Ng, then 1 is the only limit point of D. For the later use, we introduce the following definition. Definition 2.2 (i) For a 2 R, an open interval of the form (a − δ; a + δ) for some δ > 0 is called a neighbourhood of a; it is also called a δ-neighbourhood of a. (ii) By a deleted neighbourhood of a point a 2 R we mean a set of the form Dδ := fx 2 R : 0 < jx − aj < δg for some δ > 0, i.e., the set (a − δ; a + δ) n fag. With the terminologies in the above definition, we can state the followng: • A point a 2 R is a limit point of D ⊆ R if and only if every deleted neigh- bourhood of a contains at least one point of D. In particular, if D contains either a deleted neighbourhood of a or if D contains an open interval with one of its end points is a, then a is a limit point of D. Now we give a characterization of limit points in terms of convergence of se- quences. Theorem 2.1 A point a 2 R is a limit point of D ⊆ R if and only if there exists a sequence (an) in D n fag such that an ! a as n ! 1. Proof. Suppose a 2 R is a limit point of D. Then for each n 2 N, there exists an 2 D n fag such that an 2 (a − 1=n; a + 1=n). Note that that an ! a. Conversely, suppose that there exists a sequence (an) in Dnfag such that an ! a. Hence, for every δ > 0, there exists N 2 N such that an 2 (a−δ; a+δ) for all n ≥ N. In particular, for n ≥ N, an 2 (a − δ; a + δ) \ (D n fag). Exercise 2.1 Prove that a point a 2 R is a limit point of D ⊂ R if and only if there exists a sequence (an) in D such that (an) is not eventually constant and an ! a as n ! 1. [Recall that a sequence (an) is said to be eventually constant if there exists k 2 N such that an = ak for all n ≥ k.] J 2.1.2 Limit of a function f(x) as x approaches a Definition 2.3 Let f be a real valued function defined on a set D ⊆ R, and let a 2 R be a limit point of D. We say that b 2 R is a limit of f(x) as x approaches a if for every " > 0, there exists δ > 0 such that jf(x) − bj < " whenever x 2 D; 0 < jx − aj < δ; (∗) and in that case we write lim f(x) = b x!a Limit of a Function 45 or f(x) ! b as x ! a: The relations in (∗) in the above examples can also be written as x 2 D; 0 < jx − aj < δ =) jf(x) − bj < ": Exercise 2.2 Thus, lim f(x) = b if and only if for every open interval Ib containing x!a b there exists an open interval Ia containing a such that x 2 Ia \ (D n fag) =) f(x) 2 Ib: J CONVENTION: In the following, whenever we talk about limit of a function f as x approaches a 2 R, we assume that f is defined on a set D ⊆ R and a is a limit point of D. Also, when we talk about f(x), we assume that x belongs to the domain of f. For example, if we say that \f(x) has certain property P for every x in an interval I", what we mean actualy is that \f(x) has the property P for all x 2 I \ D, where D is he domain of f". Exercise 2.3 Show that, a function cannot have more than one limits. J Example 2.2 Let D be an interval and a is either in D or a is an end point of D. (i) Let f(x) = x. Since jf(x) − aj = jx − aj 8 x 2 D; it follows that for any " > 0, jf(x) − aj < " whenever 0 < jx − aj < δ := ". Hence, lim f(x) = a. x!a (ii) Let f(x) = x2 and " > 0 be given. We show that lim f(x) = a2. Note that x!a jf(x) − a2j = (jxj + jaj)jx − aj 8 x 2 D; x 6= a: Since jxj ≤ jx − aj + jaj ≤ 1 + jaj whenever jx − aj < 1, we have jf(x) − a2j = (1 + 2jaj)jx − aj 8 x 2 D; 0 < jx − aj ≤ 1: Therefore, x 2 D; 0 < jx − aj ≤ 1; (1 + 2jaj)jx − aj < " =) jf(x) − a2j < ": Thus, x 2 D; 0 < jx − aj < δ := minf1; "=(1 + 2jaj)g =) jf(x) − a2j < ": Hence, lim f(x) = a2. x!a 46 Limit, Continuity and Differentiability of Functions M.T. Nair More examples will be considered in Section 2.1.4 after proving some properties of the limit. Before that let us ask the following question. Question: Suppose f is a real valued function defined on an interval I and a 2 I. What do we mean by the statement that \lim f(x) does not exist"? x!a It means the following: For any b 2 R, there exists " > 0 such that for any δ > 0, there is atleast one xδ 2 (a − δ; a + δ) such that f(xδ) 62 (b − "; b + "). We illustrate this by a simple example. 0; −1 ≤ x ≤ 0; Example 2.3 Let f :[−1; 1] ! be defined by f(x) = We R 1; 0 < x ≤ 1: show that lim f(x) does not exist. For this let b 2 . Let us consider the following x!0 R cases: Case (i): b = 0. In this case, if 0 < " < 1, then (b − "; b + ") does not contain 1 so that f(x) 62 (b − "; b + ") for any x > 0. Case (ii): b = 1. In this case, if 0 < " < 1, then (b − "; b + ") does not contain 0 so that f(x) 62 (b − "; b + ") for any x < 0. Case (iii): b 6= 0; b 6= 1. In this case, if 0 < " < minfjbj; jb − 1jg, then (b − "; b + ") does not contain 0 and 1 so that f(x) 62 (b − "; b + ") for any x 6= 0. Thus, b is not a limit of f(x) as x approaches 0. Before going further, let us observe a property which would be used in the due course. Theorem 2.2 If lim f(x) = b, then there exists a deleted neighbourhood Dδ of a x!a and M > 0 such that jf(x)j ≤ M for all x 2 Dδ \ D. Proof. Suppose lim f(x) = b. Then there exists a deleted neighbourhood Dδ of x!a a such that jf(x) − bj < 1 for all x 2 D \ Dδ. Hence, jf(x)j ≤ jf(x) − bj + jbj < 1 + jbj 8 x 2 D \ Dδ: Thus, jf(x)j ≤ M = 1 + jbj for all x 2 Dδ \ D. 2.1.3 Limit of a function in terms of sequences Let a be a limit point of D ⊆ R and f : D ! R. Suppose lim f(x) = b. Since a is a limit point of D, we know that there exists a x!a sequence (xn) in D n fag such that xn ! a. Does f(xn) ! b? The answer is \yes". In fact, we have more! Theorem 2.3 If lim f(x) = b, then for every sequence (xn) in D such that xn ! a, x!a we have f(xn) ! b. Limit of a Function 47 Proof. Suppose lim f(x) = b. Let (xn) be a sequence in D such that xn ! a.
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