Can somebody provide some intuitive reasoning why in higher dimensions can functions in Sobolev spaces be discontinuous? It is clear in 1D why a discontinuous function cannot be in a Sobolev space (by working out), but how does that work out in higher dimensions. And, what kinds of discontinuities are allowable. Thanks, Sandy.
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You might want to check https://math.stackexchange.com/questions/749002/function-in-h1-but-not-continuous – Mansur Shakipov Oct 04 '21 at 03:40