Assume that $f_n:[0,1]\to [0,1]$ is a sequence of diffeomorphism that converges to a homeomorphism $f$ so that $$\int_0^1 \left[f'_n(x)+1/f'_n(x)\right]^2 dx<M.$$ Can we state that $f'_n$ converges up to a subsequence to $f'$ almost everywhere.
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$\begingroup$ $f$ need not be differentiable everywhere under these assumptions. $\endgroup$– Christian RemlingCommented Mar 9, 2017 at 0:42
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