The continuously differentiable functions are equipped with the topology induced by the sup norm. I know that I can make the arclength integral close to the arclength of a piecewise linear function. My idea is to take two functions which are adequately close together, and take two piecewise linear functions which are close to the integrals and show that we can make the difference of the arclength integral of the piecewise linear functions as small as we would like.
Can anyone tell me why the arclength integral is an uppersemicontinuous function on the set of continuously differentiable real-valued functions?
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real-analysis
general-topology
arc-length
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0probably Fatou's Lemma ... – 2011-10-06