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?
Example of discontinuous function having all partial derivatives
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multivariable-calculus
partial-derivative
examples-counterexamples
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0In 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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0If the partials exist and are continuous, then the function is differentiable, hence continuous at any point in the open set. – 2012-04-24