Question: Problem 58. (a) Let f: [0,1] R be a continuous function such that f(0) = f(1). Prove that there exists c [0,1/2] such that f(c)

Problem 58. (a) Let f: [0,1] R be a continuous function such that f(0) = f(1). Prove that there exists c [0,1/2] such that f(c) = f(c+1/2). (b) Let f: [0,1] R be such that for each & R that |f~'(a)| = 0 or 2. Prove that f cannot be continuous atevery x [0,1]. (c) Let f,g: [a,b] R be continuous at xg (a,b). Using the definition of continuity, prove that fg is continuous at xg

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