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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

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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.


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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.

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?


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If F is differentiable at x = a, can F '(x) be discontinuous at x = a?


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Yes!

If F is differentiable at x = a, can F '(x) be discontinuous at x = a?


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How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?


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No! discontinuities where the limits from left and right exist, but are not equal?

How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?



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Bernhard Riemann (1852, 1867) discontinuities!On the representation of a function as a trigonometric series

Defined as limit of


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Bernhard Riemann (1852, 1867) discontinuities!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


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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


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Any continuous function is integrable: be made as small as we wish by taking sufficiently small intervals.


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Bernhard Riemann (1852, 1867) be made as small as we wish by taking sufficiently small intervals. 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].


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–2 be made as small as we wish by taking sufficiently small intervals.

–1

1

2

Riemann’s function:


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At the function jumps by


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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.


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Riemann’s function: jumps by

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:


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