Introduction to Mathematical Analysis I - Second Edition

82 3.4 PROPERTIES OF CONTINUOUS FUNCTIONS Proof: The first two assertions follow from the monotonicity of f and the Intermediate Value Theorem (see also Corollary 3.4.9 ) . We will prove that f − 1 is continuous on [ c , d ] . Fix any ¯ y ∈ [ c , d ] and fix any sequence { y k } in [ c , d ] that converges to ¯ y . Let ¯ x ∈ [ a , b ] and x k ∈ [ a , b ] be such that f ( ¯ x ) = ¯ y and f ( x k ) = y k for every k . Then f − 1 ( ¯ y ) = ¯ x and f − 1 ( y k ) = x k for every k . Suppose by contradiction that { x k } does not converge to ¯ x . Then there exist ε 0 > 0 and a subsequence { x k ` } of { x k } such that | x k ` − ¯ x | ≥ ε 0 for every `. (3.7) Since the sequence { x k ` } is bounded, it has a further subsequence that converges to x 0 ∈ [ a , b ] . To simplify the notation, we will again call the new subsequence { x k ` } . Taking limits in ( 3.7 ) , we get | x 0 − ¯ x | ≥ ε 0 > 0 . (3.8) On the other hand, by the continuity of f , { f ( x k ` ) } converges to f ( x 0 ) . Since f ( x k ` ) = y k ` → ¯ y as ` → ∞ , it follows that f ( x 0 ) = ¯ y = f ( ¯ x ) . This implies x 0 = ¯ x , which contradicts ( 3.8 ) . Remark 3.4.11 A similar result holds if the domain of f is the open interval ( a , b ) with some additional considerations. If f : ( a , b ) → R is increasing and bounded, following the argument in Theorem 3.2.4 we can show that both lim x → a + f ( x ) = c and lim x → b − f ( x ) = d exist in R (see Exercise 3.2.10 ) . Using the Intermediate Value Theorem we obtain that f (( a , b )) = ( c , d ) . We can now proceed as in the previous theorem to show that f has a continuous inverse from ( c , d ) to ( a , b ) . If : ( a , b ) → R is increasing, continuous, bounded below, but not bounded above, then lim x → a + f ( x ) = c ∈ R , but lim x → b − f ( x ) = ∞ (again see Exercise 3.2.10 ) . In this case we can show using the Inter- mediate Value Theorem that f (( a , b )) = ( c , ∞ ) and we can proceed as above to prove that f has a continuous inverse from ( c , ∞ ) to ( a , b ) . The other possibilities lead to similar results. A similar theorem can be proved for strictly decreasing functions. Exercises 3.4.1 Let f : D → R be continuous at c ∈ D and let γ ∈ R . Suppose f ( c ) > γ . Prove that there exists δ > 0 such that f ( x ) > γ for every x ∈ B ( c ; δ ) ∩ D . 3.4.2 Let f , g be continuous functions on [ a , b ] . Suppose f ( a ) < g ( a ) and f ( b ) > g ( b ) . Prove that there exists x 0 ∈ ( a , b ) such that f ( x 0 ) = g ( x 0 ) . 3.4.3 Prove that the equation cos x = x has at least on solution in R . (Assume known that the function cos x is continuous.) 3.4.4 Prove that the equation x 2 − 2 = cos ( x + 1 ) has at least two real solutions. (Assume known that the function cos x is continuous.) 3.4.5 Let f : [ a , b ] → [ a , b ] be a continuous function.