# Are geodesics locally minimizing in continuous curves?

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?

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Continuous curves don't neccessarily have a length. Let's add the hypothesis that the curve be rectifiable, also lets assume we are in a complete Riemannian manifold, then the answer is yes. – Charlie Frohman Jun 27 '11 at 12:43
I'd think a non-rectifiable continuous curve could naturally be assigned infinite "length" (by using the same supremum of distances as in the definition for rectifiable curves), so that you wouldn't have to worry about them as candidates for minimizing lengths. – Andreas Blass Jun 27 '11 at 13:09
To amplify @Charlie's comment: how do you define the length of a curve? If as the limit of piecewise-smooth approximations, then the statement is immediate. If not, you should tell us your definition of length... – Igor Rivin Jun 27 '11 at 15:31