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.