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Construct an element of $ W^{1,2}(\mathbb{R}^2)$ that is unbounded at the origin. I want to use this to show that $ W^{1,2}(\mathbb{R}^2)$ is not contained in $ C(\mathbb{R}^2)$ . (This relates to $ W^{1,2}(\mathbb{R})$ being a “borderline” case for Sobolev embedding.) My attempt: I’ve been looking at the function $ u(x) = \log\log(1+\frac{1}{|x|})$ because IRead more