A continuous, nowhere differentiable but invertible function?

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.

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.)

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