Theorem 7.16 in the coarea inequality (B. Esmayli, P. Hajlasz) arxiv.org/pdf/2006.00419.pdf has an inequality with a non-integer exponents. Might be of interest.

Any update on this? We have proven a coarea inequality for monotone Sobolev functions, so this could seriously be related.

Bc continuous and positive means positive in a whole interval. Then MVT kicks in. I was asked this by some friends who were my seniors and taking analysis 3 while I was in analysis 1. I did find the counter-example on my own. One little math discovery I was proud of :)

I would begin by saying that "a dense subspace is just the full space in the first place". :((

Once you have a Lipschitz map into part of your rectifiable set, you can ignore the set where the (metric) derivative is non-injective because by a Sard-type theorem the image of this set has zero Hausdorff measure. On the rest of the domain the map is locally one-to-one, and indeed bi-Lipschitz because locally you have positive lower bound on modulus of derivative. (By metric derivative I am talking about Kirchheim's differentiability result from 1994.)

@PiotrHajlasz What a nice Prelim question!