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

1 / 21

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

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Riemann’s example of function f for which exists for all x , but is not' - preston

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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?

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?

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: