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# hw5 - f D → R is continuous Note This is book problem 1 2...

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MATH 410: Homework 5 Due in class Friday 3/2/2012 Section 3.3 1. Prove that there is a solution to the equation * 1 x + x 2 + x 2 - 2 x = 0 with x > 0 Note: This is book problem 3. 2. Suppose that f : R R is continuous and that f ( R ) is bounded. Prove that there is a solution ** to the equation f ( x ) = x with x R . Note: This is book problem 6. Section 3.4 1. For each of the following statements, determine if true or false. If false provide a counterex- ample. True statements need no justification. (a) Every continuous function f : R R is uniformly continuous. * (b) Every continuous function f : [0 , 1) R is uniformly continuous. * (c) Every continuous function f : [0 , 1] R is uniformly continuous. * (d) Every uniformly continuous function f :
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Unformatted text preview: f : D → R is continuous. * Note: This is book problem 1. 2. Prove that f : R → R given by f ( x ) = x 3 is not uniformly continuous. * Note: This is book problem 5. 3. Suppose that f : ( a, b ) → R is uniformly continuous. prove that f is bounded. ** Note: This is book problem 10. Section 3.5 1. De±ne f ( x ) = √ x for x ≥ 0. Verify the ǫ-δ criterion for continuity at x = 4. ** Note: This is part of book problem 2. There’s a hint in the book. 2. De±ne * f ( x ) = b x + 1 if x ≤ 3 4 2 if x > 3 4 Use the ǫ-δ criterion to show that f is not continuous at x = 3 4 . Note: This is book problem 4....
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