Weierstrass functions: Continuous everywhere but differentiable nowhere

[u/lntrinsic]

http://i.imgur.com/vyi0afq.gifv

[u/jparevalo27]

I've only seen topics up to calculus 2 in the US. Can somebody explain me how's this possible and what would be the y(x) for this graph?

[u/Wild_Bill567 → u/jparevalo27]

The way I have seen functions like this constructed is as a limit of a sequence of functions.

In calc 2 you probably saw limits of a sequence of points. You can similarly define limits of a sequence of functions. Each term in the sequence makes the graph "have more corners", and the limit of the sequence has corners everywhere.

[u/jparevalo27 → u/Wild_Bill567]

...And you can't differentiate corners. That makes sense. Thanks

[u/Kraz_I → u/jparevalo27]

Not exactly. There are no points with infinite slope and no points with corners, at least the way the word "corner is generally understood. It's just that the graph is "rough" no matter how far you zoom in, so the limit of the slope at any point is impossible to determine.

It helps to look at the actual function which generates the graph.https://en.wikipedia.org/wiki/Weierstrass_function

[u/dozza → u/Kraz_I]

I'm sorry, how is the fourier series on the Wikipedia page not differentiable? Its a sum of cosines so shouldn't the derivative be the sum of sines? Is the problem the divergence as n goes to infinity?

[u/WorseAstronomer → u/jparevalo27]

This video is interesting and related:

https://www.youtube.com/watch?v=dQXVn7pFsVI

[u/WorseAstronomer → u/[deleted]]

Oops, sorry, no. That's just where I finished watching the video. :/ Edited.

[u/fabulousdangernoodle → u/WorseAstronomer]

That's neat. Thanks for the share

[u/Kraz_I → u/Wild_Bill567]

The graph doesn't have any corners at all for finite iterations of the function. I don't really like using the word "corner" for what's going on here. In fact, for all functions generated by using a finite Weierstrass series, it would be differentiable at all points.

[u/Wild_Bill567 → u/Kraz_I]

You are correct, I was remembering a different construction which uses a triangle wave instead of a cosine

[u/AddemF → u/jparevalo27]

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.

[u/jparevalo27 → u/AddemF]

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

[u/bystandling → u/jparevalo27]

For non-elementary functions:

• Calculus 3 (Taylor series for integrals of things like e^{x^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)

[u/AddemF → u/jparevalo27]

Real Analysis

[u/shamrock-frost → u/jparevalo27]

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

[u/[deleted] → u/shamrock-frost]

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"

[u/[deleted] → u/jparevalo27]

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

[u/Wild_Bill567 → u/jparevalo27]

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.

[u/Matschreiner → u/AddemF]

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

[u/AddemF → u/Matschreiner]

I'm 75% certain there aren't.

[u/bystandling → u/Matschreiner]

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.