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?