I have a geometry project coming up, and have stumbled on the following problem, wondering if anybody had some help to offer.
Let $\gamma:[0,1]\rightarrow \mathbb{R}^2$ be a rectifiable curve. I.e the following value is finite
$$L(\gamma)=sup_{\sigma\in P}\sum_{i=1}^{n}||\gamma(x_{i+1})-\gamma(x_i)||$$
Where P is the set of subdivisions of the unit interval.
I want to show (or find a counterexample) that then $\gamma$ is locally convex or concace, i.e $\forall x\in [0,1]$ there exists some $\delta>0$ such that $\gamma|_{[x-\delta, x+\delta]}$ is either convex or concave.