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Riemann’s example of function f for which exists for all x , but is not

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## Riemann’s example of function f for which exists for all x , but is not

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**Riemann’s example of function f for which**exists for all x, but is not differentiable when x is a rational number with even denominator.**Riemann’s example of function f for which**exists for all x, but is not differentiable when x is a rational number with even denominator. What does a derivative look like? Can we find a function that can’t be a derivative but which can be integrated? Does a derivative have to be continuous?**If F is differentiable at x = a, can F '(x) be**discontinuous at x = a?**Yes!**If F is differentiable at x = a, can F '(x) be discontinuous at x = a?**How discontinuous can a derivative be? Can it have jump**discontinuities where the limits from left and right exist, but are not equal?**No!**How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?**The derivative of a function cannot have any jump**discontinuities!**Bernhard Riemann (1852, 1867) On the representation of a**function as a trigonometric series Defined as limit of**Bernhard Riemann (1852, 1867) On the representation of a**function as a trigonometric series Defined as limit of Key to convergence: on each interval, look at the variation of the function**Integral exists if and only if can**be made as small as we wish by taking sufficiently small intervals. Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series Defined as limit of Key to convergence: on each interval, look at the variation of the function**Bernhard Riemann (1852, 1867) On the representation of a**function as a trigonometric series Riemann gave an example of a function that has a jump discontinuity in every subinterval of [0,1], but which can be integrated over the interval [0,1].**–2**–1 1 2 Riemann’s function:**At the function**jumps by**At the function**jumps by Riemann’s function: The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number.**Riemann’s function:**At the function jumps by The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number. Conclusion: