# The Weierstrass function's' odd qualities

• I
I recently stumbled on the Weierstrass function, whose main claim to fame (as I understand it) is to be continuous everywhere, but non-differentiable everywhere as well. Apparently I was in good company with Gauss' and others who assumed that to be impossible!

I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.

.Scott
Homework Helper
I recently stumbled on the Weierstrass function, whose main claim to fame (as I understand it) is to be continuous everywhere, but non-differentiable everywhere as well. Apparently I was in good company with Gauss' and others who assumed that to be impossible!

I guess I'm asking, is this dichotomy mostly down to the loosened definition of continuity, i.e. Hölder continuous instead of Lipschitz continuous? Not that I dispute the validity or use of the former definition, but it certainly would mean that my intuition was correct in the stricter Lipschitz sense.
No, it is continuous in the true sense of the term. But what is d(abs(x))/dx when x=0? The problem with the Weierstrass function is that it has that problem for all x.

mfb
Mentor
According to this discussion, it is Hölder continuous for every ##0<\alpha<1##, but not Lipschitz continuous (which would be ##\alpha=1##).
These are stricter criteria than the regular continuity, however. The function is continuous.

S.G. Janssens
No, it is continuous in the true sense of the term. But what is d(abs(x))/dx when x=0? The problem with the Weierstrass function is that it has that problem for all x.

Oooh, that is an excellent explanation, thanks a lot. It's been a long time since college math, I had forgotten that a "kink" in a graph is non-differentiable. From there it's not too hard to construct an "only kinks" function.

S.G. Janssens