Introduction to Mathematical Analysis I - Second Edition

74 3.3 CONTINUITY 3.2.7 Find each of the following limits if they exist: (a) lim x → 1 + x + 1 x − 1 . (b) lim x → 0 + x 3 sin ( 1 / x ) . (c) lim x → 1 ( x − [ x ]) . 3.2.8 For a ∈ R , let f be the function given by f ( x ) = ( x 2 , if x > 1; ax − 1 , if x ≤ 1 . Find the value of a such that lim x → 1 f ( x ) exists. 3.2.9 Determine all values of ¯ x such that the limit lim x → ¯ x ( 1 + x − [ x ]) exists. 3.2.10 Let a , b ∈ R and suppose f : ( a , b ) → R is increasing. Prove the following. (a) If f is bounded above, then lim x → b − f ( x ) exists and is a real number. (b) If f is not bounded above, then lim x → b − f ( x ) = ∞ . State and prove analogous results in case f is bounded below and in case that the domain of f is one of ( − ∞ , b ) , ( a , ∞ ) , or ( − ∞ , ∞ ) . 3.3 CONTINUITY Definition 3.3.1 Let D be a nonempty subset of R and let f : D → R be a function. The function f is said to be continuous at x 0 ∈ D if for any real number ε > 0, there exists δ > 0 such that if x ∈ D and | x − x 0 | < δ , then | f ( x ) − f ( x 0 ) | < ε . If f is continuous at every point x ∈ D , we say that f is continuous on D (or just continuous if no confusion occurs). Figure 3.1: Definition of continuity.