## Continuous nowhere differentiable functions

Following a theme from two years ago, we will have a final project for this course, due Wednesday, December 18, by noon, but feel free (and encouraged) to turn it in earlier. (As discussed in lecture, the project is voluntary for some of you. Contact me if you are not sure whether it is required or voluntary for you.)

There are many excellent sources on the topic of continuous nowhere differentiable functions. Johan Thim’s Master thesis, written under the supervision of Lech Maligranda, is available online, here, but feel free to use any other sources you find relevant.

Please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write (better yet, type) a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Though not required, feel free (and encouraged) to add additional information you consider relevant for context.

(For an example of what I mean by relevant additional information: Weierstrass function is $\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$ where $0, $b$ is an odd positive integer, and $\displaystyle ab>1+\frac32\pi$. It may be interesting to add a discussion of precisely what conditions are needed from $a,b$ to ensure (continuity and) nowhere differentiability; Weierstrass original requirements are more restrictive than necessary. For another example, Schoenberg functions, discussed in Thim’s thesis, give a natural example of a space filling curve, so consider including a proof of this fact.)

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute. At the end, it would be nice to make at least some of the notes available online, please let me know when you turn it in whether you grant me permission to host your note on this blog.

Here is a list of the projects I posted on the blog, from last time:

Contact me (by email) as soon as you have chosen the example you will work on, to avoid repetitions; I will add your name and the chosen example to the list below as I hear from you.

List of projects:

• Joe Busick: Katsuura function.
• Paul Carnig: Darboux function.
• Joshua Meier: A variant of Koch’s snowflake.
• Paul Plummer: Lynch function.
• Veronica Schmidt: McCarthy function.

### 2 Responses to Continuous nowhere differentiable functions

1. […] and nowhere differentiable can be found in Thim’s master’s thesis mentioned in the previous post. Weierstrass proof was published by DuBois-Reymond in […]

2. […] assigning a final project on the topic of continuous nowhere differentiable functions (see here and here for the previous […]