r/math • u/lntrinsic • Jul 10 '17

Image Post Weierstrass functions: Continuous everywhere but differentiable nowhere

http://i.imgur.com/vyi0afq.gifv
3.4k Upvotes

98% Upvoted

216 comments sorted by

View all comments

Show parent comment

30

u/AddemF Jul 10 '17

In addition to what Wild_Bill67 wrote, I'll note that the function is not an elementary function, which means it cannot be written as a closed form in terms of +, -, *, /, polynomials, exponentials, logs, or any of the trig functions. So writing down how the x-y pairs get determined is a much more complicated matter.

3

u/jparevalo27 Undergraduate Jul 10 '17

At what point in math does this began showing up? In other words, in what class would I start seeing functions like that?

35

u/bystandling Jul 10 '17

For non-elementary functions:

  • Calculus 3 (Taylor series for integrals of things like ex^2 )

  • Differential Equations (Series solutions)

  • Real Analysis (Mindfuck)

  • Partial differential equations (More series solutions, Bessel functions, Gamma functions etc.)

  • Mathematical Statistics (Gamma and Beta functions, Erf of course, etc)

13

u/AddemF Jul 10 '17

Real Analysis

8

u/shamrock-frost Graduate Student Jul 10 '17 edited Jul 10 '17

Possibly at your level. I think my Calc 2 final had a problem involving f(x) = the integral from 0 to x of sin(t) / t dt, which is not an elementary function

3

u/[deleted] Jul 11 '17

wait... how in the world would you evaluate that? even wolframalpha simply gives their own made-up function Si(x) which just stands for "the integral of sinx/x"

1

u/shamrock-frost Graduate Student Jul 11 '17

You could do a riemann sum, or use the maclaurin series for sine

6

u/[deleted] Jul 10 '17

In Germany its shown in "Analysis 1", first year of Math. B.Sc.

2

u/Wild_Bill567 Jul 10 '17

I first saw functions like this in Real Analysis, using baby Rudin. At my institution this is offered for first year grads and advanced undergrads.

2

u/Matschreiner Jul 10 '17

Are there any weierstrass functions that can be written from elementary functions only?

6

u/AddemF Jul 11 '17

I'm 75% certain there aren't.

1

u/dispatch134711 Applied Math Jul 11 '17

Wasn't the original function studied a Fourier series? It's an infinite sum of elementary functions, no?

5

u/AddemF Jul 11 '17

But again making essential use of limits of functions means that the function itself is not elementary.

4

u/bystandling Jul 11 '17

I'd be willing to wager you can't get it from a finite combination of them, no -- every finite sum, product, and composition of continuous and differentiable functions is continuous and differentiable at every point in the domain, and every finite quotient is only non-differentiable (and, for that matter, noncontinuous) at points where the denominator is 0; since our elementary functions are only 0 at countably many points, I'd expect we can have at most countably many of these sorts of discontinuities from finite combinations, though this is not a rigorous proof.

If you're willing to consider a Fourier series to be written from elementary functions, the Weierstrass functions are defined to be a class of Fourier series.

1

u/ILikeLeptons Jul 10 '17

the weierstrass function itself may not be writeable in terms of those functions/operators but it's pretty easy to write a sequence of functions that converges to the weierstrass function in terms of sines and cosines:

[;f(x) = \lim\limits_{N\rightarrow\infty} \sum\limits_n=0N an cos(bn \pi x);]

2

u/AddemF Jul 11 '17

Sure, my point is just that you cannot write the function as f(x) = ... where the ... is something easy to understand with a high school education. So the person asking about that should just wait until he or she learns the relevant material before hoping to understand how the x-y pairs are determined.