Introduction to Mathematical Analysis I - Second Edition

88 3.5 UNIFORM CONTINUITY We now prove a result that characterizes uniform continuity on open bounded intervals. We first make the observation that if f : D → R is uniformly continuous on D and A ⊂ D , then f is uniformly continuous on A . More precisely, the restriction f | A : A → R is uniformly continuous on A (see Section 1.2 for the notation). This follows by noting that if | f ( u ) − f ( v ) | < ε whenever u , v ∈ D with | u − v | < δ , then we also have | f ( u ) − f ( v ) | < ε when we restrict u , v to be in A . Theorem 3.5.5 Let a , b ∈ R and a < b . A function f : ( a , b ) → R is uniformly continuous if and only if f can be extended to a continuous function ˜ f : [ a , b ] → R (that is, there is a continuous function ˜ f : [ a , b ] → R such that f = ˜ f | ( a , b ) ). Proof: Suppose first that there exists a continuous function ˜ f : [ a , b ] → R such that f = ˜ f | ( a , b ) . By Theorem 3.5.4 , the function ˜ f is uniformly continuous on [ a , b ] . Therefore, it follows from our early observation that f is uniformly continuous on ( a , b ) . For the converse, suppose f : ( a , b ) → R is uniformly continuous. We will show first that lim x → a + f ( x ) exists. Note that the one sided limit corresponds to the limit in Theorem 3.2.2 . We will check that the ε - δ condition of Theorem 3.2.2 holds. Let ε > 0. Choose δ 0 > 0 so that | f ( u ) − f ( v ) | < ε whenever u , v ∈ ( a , b ) and | u − v | < δ 0 . Set δ = δ 0 / 2. Then, if u , v ∈ ( a , b ) , | u − a | < δ , and | v − a | < δ we have | u − v | ≤ | u − a | + | a − v | < δ + δ = δ 0 and, hence, | f ( u ) − f ( v ) | < ε . We can now invoke Theorem 3.2.2 to conclude lim x → a + f ( x ) exists. In a similar way we can show that lim x → b − f ( x ) exists. Now define, ˜ f : [ a , b ] → R by ˜ f ( x ) =   f ( x ) , if x ∈ ( a , b ) ; lim x → a + f ( x ) , if x = a ; lim x → b − f ( x ) , if x = b . By its definition ˜ f | ( a , b ) = f and, so, ˜ f is continuous at every x ∈ ( a , b ) . Moreover, lim x → a + ˜ f ( x ) = lim x → a + f ( x ) = ˜ f ( a ) and lim x → b − ˜ f ( x ) = lim x → b − f ( x ) = ˜ f ( b ) , so ˜ f is also continuous at a and b by Theorem 3.3.2 . Thus ˜ f is the desired continuous extension of f . Exercises 3.5.1 Prove that each of the following functions is uniformly continuous on the given domain: (a) f ( x ) = ax + b , a , b ∈ R , on R . (b) f ( x ) = 1 / x on [ a , ∞ ) , where a > 0. 3.5.2 I Prove that each of the following functions is not uniformly continuous on the given domain: (a) f ( x ) = x 2 on R . (b) f ( x ) = sin 1 x on ( 0 , 1 ) . (c) f ( x ) = ln ( x ) on ( 0 , ∞ ) . 3.5.3 Determine which of the following functions are uniformly continuous on the given domains.