Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

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Twice continuously differentiable function $f:\Bbb R^2\to \Bbb R$ such that $f, f_{xx}, f_{yy}\in L^2$ is bounded

Suppose $f:\Bbb R^2\to \Bbb R$ is twice continuously differentiable, and that $f, f_{xx}, f_{yy}\in L^2$. Then is it true that $f$ is bounded? I first tried to find a counterexample, but I couldn't. ...