There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you consider strange and especially like.


Please also give a reason why you find this function strange and why you like it. Perhaps you could also give some kind of reference where to find further information.


As usually: Please only mention one function per post - and let the votes decide :-)



What is your favorite "strange" function?
A **[Brownian motion sample path](http://en.wikipedia.org/wiki/Wiener_process#Some_properties_of_sample_paths)**.


These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are nowhere differentiable, have unbounded variation, attain local maxima and minima in every interval... Many, many papers and books have been written about their strange properties.


Edit: As commented, I should clarify the term "sample path". Brownian motion is a stochastic process $B\_t$. We say a sample path of Brownian motion has some property if the function $t \mapsto B\_t$ has that property almost surely. So, run a Brownian motion, and with probability 1 you will get a function with all these weird properties.


