Are there any functions that are (always) continuous yet not differentiable? Or vice-versa?

It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable?

The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But are there any that do not satisfy this?

• +1: I think this is a good on-topic question if this site is to be useful to undergraduate math majors. The Weierstrass function has been very nicely identified in the answers below, and it is an important counter-example that comes up immediately in advanced calculus. – Tom Stephens Jul 21 '10 at 3:50

It's easy to find a function which is continuous but not differentiable at a single point, e.g. f(x) = |x| is continuous but not differentiable at 0.

Moreover, there are functions which are continuous but nowhere differentiable, such as the Weierstrass function.

On the other hand, continuity follows from differentiability, so there are no differentiable functions which aren't also continuous. If a function is differentiable at $$x$$, then the limit $$(f(x+h)-f(x))/h$$ must exist (and be finite) as $$h$$ tends to 0, which means $$f(x+h)$$ must tend to $$f(x)$$ as $$h$$ tends to 0, which means $$f$$ is continuous at $$x$$.

• Moreover, the set of continuous functions which are nowhere differentiable is residual, so one can prove their existence without actually constructing an example. – Akhil Mathew Jul 20 '10 at 23:06
• @Akhil Mathew: on the other hand, from the proof, via Baire's theorem, that the set of functions with one differentiability point is meager we can construct an example of a nowhere differentiable function (e.g. a series of see-saw functions with appropriate slopes running off to infinity). – G. Rodrigues Feb 23 '11 at 14:45
• @AkhilMathew, but unbelievers won't believe your "pure existence proof"... – vonbrand Feb 23 '13 at 21:20

Actually, in some sense, almost all of the continuous functions are nowhere differentiable: http://en.wikipedia.org/wiki/Weierstrass_function#Density_of_nowhere-differentiable_functions

A natural class of examples would be paths of Brownian motion. These are continuous but non-differentiable everywhere.

You may also be interested in fractal curves such as the Takagi function, which is also continuous but nowhere differentiable. (I think Wikipedia calls it the "Blancmange curve".) I like this one better than the Weierstrass function, but this is personal preference.

Brownian Motion

Takagi function

I really like this answer I went over in my lecture: 