Example of length structure with Euclidian intrinsic metric but different path lengths This is an exercise from "A course in metric geometry" by Burago and Ivanov I am having some troubles with.
Give an example of length structure on the plane for which all continuous curves are admissible, the resulting intrinsic metric is the standard Euclidian one, but lengths of some curves differ from their euclidian lengths. 
I have an example for $C^2$ paths but struggling to do this for continuous curves. Thank you.
 A: A hint: Define the length $L(c)$ of every piecewise-linear path $c$ to be its Euclidean length. Define the length $L(c)$ to be $\infty$ for all other continuous paths $c$. Now, go through the list of axioms of a length structure (see e.g.here) that this is indeed a length structure. Lastly, think  of a path $c$ such that $L(c)$ differs from the Euclidean length of $c$. 
Edit. Suppose, in addition to the usual axioms of a length space you want the set of paths of finite length to be the same as in the Euclidean case. Here is a modification of my construction above:
Let $A$ be the subset of rectifiable (from Eucldiean viewpoint) continuous curves (maps from finite intervals to $R^2$). Let's define a length structure on $A$ as follows. For each $f:[a,b]\to R^2$ which belongs to $A$, let $S_f\subset [a,b]$ be the union of all open  intervals $I\subset [a,b]$ such that $f$ restricted to these intervals is smooth (say, $C^1$). Then the Euclidean length $L(f)$ of $f$ equals 
$$
L(f)=\int_{S_f}|f'(t)|dt + R_f.
$$
Furthermore, let $G_f$ be the union of all open subintervals of $S_f$ such that the restriction of $f$ to each of these subintervals is linear (more precisely, affine). Lastly, let $H_f:= S_f - G_f$. Now, set
$$
L'(f):= \int_{G_f}|f'(t)|dt + 2\int_{H_f}|f'(t)|dt + R_f. 
$$
Hence, for each piecewise-linear path $f$, $L'(f)=L(f)$ but, in general, $L'(f)\ne L(f)$. For each nonrectifiable path $f$ set $L'(f)=\infty$. 
I will leave you to check that $L'$ satisfies all the axioms of a length structure and defines the usual (Euclidean) distance on $R^2$ (since we did not change lengths of linear paths).  
