A continuous, nowhere differentiable but invertible function?

I am aware of a few example of continuous, nowhere differentiable functions. The most famous is perhaps the Weierstrass functions

$$W(t)=\sum_k^{\infty} a^k\cos\left(b^k t\right)$$

but there are other examples, like the van der Waerden functions, or the Faber functions. Most of these "look like" some variation of:

(Weierstrass functions from Wolfram)

Specifically, they are clearly not invertible. Since these functions are generally self-similar at many scales, this non-invertability would seem to hold essentially everywhere.

I'm wondering if it's possible to construct such a function which is invertible. Intuitively, maybe this would be "jittery" in the same way as the Weierstrass function, but if it had a slope which always increased, it would be invertible. Or perhaps there is at least an example in which the function is invertible over some segment of the range.

Solution 1:

Interestingly, there are no such examples! For a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ to be invertible, it must be either monotone increasing or decreasing. A famous classical result in analysis, Lebesgue's Monotone Function Theorem, states that any monotone function on an open interval is differentiable almost everywhere. Hence, there are no continuous functions that are invertible and nowhere differentiable.

Solution 2:

Invertible implies bijective by set theory, and bijective together with continuity implies strictly increasing or decreasing, which imply differentiability almost everywhere! (This is known as Lebesgue's monotone function theorem.)

Solution 3:

If $f:(a,b)\to\Bbb R$ is continuous and injective it must be monotone, hence differentiable almost everywhere.

Solution 4:

I will assume by invertible you mean injective, i.e. bijective on its image. As others have mentioned, there are no such examples among functions from $\mathbb{R}$ to $\mathbb{R}$.

However you might find such examples in other topological spaces.

An example $\mathbb{R} \rightarrow \mathbb{R}^2$ is given by $g(t) = (W(t), t)$ with $W$ the Weierstrass function; it is obviously injective and continuous but nowhere differentiable, considering its projection to the first coordinate.

You can also build an example from $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ by considering $g(t_1, t_2) = (t_1, W(t_1) + t_2)$ or its inverse, given by $g^{-1}(x_1, x_2) = (x_1, x_2 - W(x_1))$.

An injection $\mathbb{R}^2 \rightarrow \mathbb{R}$ cannot be continuous so you won't find a counterexample there.