Last night you had the strangest dream. Among other things, a guy named Dirac appeared in...

  

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Last night you had the strangest dream. Among other things, a guy named Dirac appeared in your dreams, and he started talking about the importance of the Heaviside function. The Heaviside function, denoted by H(x), is given by H(x) = == {{ 0 if 10 x 0 if 0 < x <10 a) Use the FTC to prove that the following statement is false: there exists a continuous function f on [-10, 10] such that H(x) = f(t)dt. b) Let us see what happens when we allow ourselves to choose f from a larger class of functions. Decide whether the following statement is true or false, and prove it or disprove it accordingly: there exists a bounded function f, integrable on [-10, 10] such that H(x) = f(t)dt. Last night you had the strangest dream. Among other things, a guy named Dirac appeared in your dreams, and he started talking about the importance of the Heaviside function. The Heaviside function, denoted by H(x), is given by H(x) = == {{ 0 if 10 x 0 if 0 < x <10 a) Use the FTC to prove that the following statement is false: there exists a continuous function f on [-10, 10] such that H(x) = f(t)dt. b) Let us see what happens when we allow ourselves to choose f from a larger class of functions. Decide whether the following statement is true or false, and prove it or disprove it accordingly: there exists a bounded function f, integrable on [-10, 10] such that H(x) = f(t)dt.