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Is it possible to a real-valued function of two variables defined on an open set to have partial derivatives of all order and to be discontinuous at some point or maybe at each point?

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    In order that $\partial f/\partial x$ and $\partial f/\partial y$ be defined, $f$ would have to be continuous as a function of $x$ with $y$ fixed and vice-versa, so the discontinuity would have to happen as one approaches a point from a diagonal direction.2012-04-24
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    If the partials exist and are continuous, then the function is differentiable, hence continuous at any point in the open set.2012-04-24

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