Question

PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which is again reasonable. The purpose of this problem is to provide a counterexample to both of these beliefs. Throughout this problem, let f :RR be the function defined by f(x)- This function is often called the indicator function of Q, for obvious reasons. (1) Describe how you might graph this function. Describe in words the apparent difficulty in doing so. (2) Prove that f is discontinuous at every point in its domain Hint: Use the characterization of continuity in terms of limits. To prove that f is discontinuous at a particular point a є R, you < zn-1) be a partition of [0,1]. Prove that U(f, P)-1 and (3) Let P-(0-zo < xi < L(f, P) 0. Hint: Use the conclusions of Problem 1 (4) Prove thatofdz does not exist.

Answer 1

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