> > There are probably even good examples of functions that are not differentiable on any open interval but are everywhere continuous, but I can't think of any right now.
>
> You can find loads of them once you go into the complex plane. I haven't looked into this, but I have heard that you can get functions that consist entirely of corners, so are continuous everywhere but differentiable nowhere. I think we cover complex Calculus next year, so I'll keep you posted. There's nothing in any of my books on it, unfortunately.
>
> Matthew

The thing about that is that it's surprisingly easy to define non-differentiable complex-valued functions, since the differentiability requirement is even stricter than requiring the existence of first partial derivatives (in fact, I think you need to have partial derivatives of *all* orders).

A quick example of a complex-valued function which is continuous everywhere but differentiable nowhere is f(z) = conjugate of z. In other words, if x and y are the real and imaginary parts, f(x + iy) = x - iy. It's differentiable nowhere because the Cauchy-Riemann equations fail to hold anywhere (see link).

But since you mentioned corners, I think you're talking about real-valued functions of a complex variable (my example was a complex-valued function of a complex variable), which you could think of as just a function of two real variables. In that case, the differentiability (oooh, a 17-letter word) requirement isn't as strong, so you have to come up with more interesting examples.

I'm pretty sure you can manage to do it with just real-valued functions of a single real variable though, but I only *heard* you could construct one; I didn't actually ever see an example.