Introduction to Mathematical Analysis I - Second Edition

78 3.4 PROPERTIES OF CONTINUOUS FUNCTIONS Definition 3.4.1 We say that the function f : D → R has an absolute minimum at ¯ x ∈ D if f ( x ) ≥ f ( ¯ x ) for every x ∈ D . Similarly, we say that f has an absolute maximum at ¯ x if f ( x ) ≤ f ( ¯ x ) for every x ∈ D . Figure 3.2: Absolute maximum and absolute minimum of f on [ a , b ] . Theorem 3.4.2 — Extreme Value Theorem. Suppose f : D → R is continuous and D is a compact set. Then f has an absolute minimum and an absolute maximum on D . Proof: Since D is compact, A = f ( D ) is closed and bounded (see Theorem 2.6.5 ) . Let m = inf A = inf x ∈ D f ( x ) . In particular, m ∈ R . For every n ∈ N , there exists a n ∈ A such that m ≤ a n < m + 1 / n . For each n , since a n ∈ A = f ( D ) , there exists x n ∈ D such that a n = f ( x n ) and, hence, m ≤ f ( x n ) < m + 1 / n . By the compactness of D , there exists an element ¯ x ∈ D and a subsequence { x n k } that converges to ¯ x ∈ D as k → ∞ . Because m ≤ f ( x n k ) < m + 1 n k for every k , by the squeeze theorem (Theorem 2.1.6 ) we conclude lim k → ∞ f ( x n k ) = m . On the other hand, by continuity we have lim k → ∞ f ( x n k ) = f ( ¯ x ) . We conclude that f ( ¯ x ) = m ≤ f ( x ) for every x ∈ D . Thus, f has an absolute minimum at ¯ x . The proof is similar for the case of absolute maximum. Remark 3.4.3 The proof of Theorem 3.4.2 can be shortened by applying Theorem 2.6.4 . However, we have provided a direct proof instead. Corollary 3.4.4 If f : [ a , b ] → R is continuous, then it has an absolute minimum and an absolute maximum on [ a , b ] .