In every lecture on Riemannian geometry it is standard to prove that geodesic curves are locally length minimizing.
The only thing I find confusing about this is, that here length minimizing means: compared to all piecewise smooth curves
in contrast to, say, all continuous curves. So my question is:


Are geodesics locally length minimizing in the continuous curves?


If generally they are not: Under which conditions can we obtain such a result? Can you give any counterexamples?



Are geodesics locally minimizing in continuous curves?
Your question will be trivial once you give a definition of the length of curve in a Riemannian manifold.


For example, you may define distance as infimum of lengths piecewise smooth curves connecting given points.
Then you define length of general curve as you do it in a metric space...


