f07HW2 - ). ii) Prove that f is uniformly continuous on [1...

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MAT 125A Homework 2 M.Fukuda Please submit your answers at the discussion session on October 18th Thursday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. i) Suppose we have a continuous function f : [ a, b ] R such that f ( a ) + f ( b ) = 0. Show that f ( x ) = 0 for some x [ a, b ]. ii) Suppose we have a polynomial of degree n : p ( x ) = a 0 + a 1 x + a 2 x 2 + . . . + a n x n , where n is an odd number and a n n = 0. Show that this polynomial p ( x ) has a real root. You can assume that polynomials are continuous. Hint: Consider how the value p ( x ) behaves when x is a large negative/positive number. 2. Let f be a function on (0 , + ): f ( x ) = 1 x . i) Prove that f is not uniformly continuous on (0 , +
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Unformatted text preview: ). ii) Prove that f is uniformly continuous on [1 , 2] by assuming the fact that f is continuous on (0 , + ). iii) Prove that f is uniformly continuous on [1 , + ) by using the property in Denition 19.1. 3. i) Let f, g be uniformly continuous functions on ( a, b ). Show that the function f + g is uniformly continuous on ( a, b ) by using the property in Denition 19.1. ii) Let f be a function on ( a, b ). Suppose that f is uniformly continuous on ( a, c ] and [ c, b ) for some c such that a < c < b . Then, Prove that f is uniformly continuous on ( a, b ) by using the property in Denition 19.1. 1...
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