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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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Presentation Transcript
slide1
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.

slide2
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?

slide4
Yes!

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

slide9
How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal?
slide10
No!

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

slide13
Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series

Defined as limit of

slide14
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

slide15
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

slide17
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].

slide18
–2

–1

1

2

Riemann’s function:

slide19
At the function jumps by
slide20
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.

slide21
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:

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