SCHWEITZER 1980 Problem 6

Let us call a continuous function $f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $f(t)=f(h(t))$ is satisfied for every $\alpha \leq t \leq \beta$ ); otherwise $f$ is irreducible. Construct irreducible $f : [a,b] \rightarrow \mathbb{R}^2$ and $g : [c,d] \rightarrow \mathbb{R}^2$ such that $f([a,b])=g([c,d])$ and

(a) both $f$ and $g$ are rectifiable but their lengths are different;

(b) $f$ is rectifiable but $g$ is not.

A. Csaszar