Question: S Let . ;F/ D .R;B/ and X W ! R be an arbitrary real function. Verify the following: (a) If X is piecewise monotone

S Let . ;F/ D .R;B/ and X W ! R be an arbitrary real function. Verify the following:

(a) If X is piecewise monotone (i.e., R may be decomposed into at most countably many intervals, on each of which X is either increasing or decreasing), then X is a random variable.

(b) If X is differentiable with (not necessarily continuous) derivative X0, then X0 is a random variable.

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