Suppose $f(x, 0) = 0$ for all $x$ in some domain of definition. Let $g = \partial f/\partial x$. Does it follow that $g(x, y) = 0$ for all $(x, y)$ in our domain?
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No, for example: $$f(x,y)=xy$$ $$f(x,0)=0$$ $$g(x,y)=y$$ |
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