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Let $f:R^2 \rightarrow R$ and $a$ in $R^2$. If we know one partial derivative exists in an open ball around $a$ and is continuous at $a$ and that the second partial derivative exists at $a$, show that $f$ is differentiable at $a$.

I've been messing around with the definition of continuous and differentiable for a while now and haven't been able to get very far. Some hints would be much appreciated!

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Try to write your problem as $$f(x,y) = (f(x,y)-f(x,0))+(f(x,0)-f(0,0))+f(0,0).$$ For the first difference use the continuity of the partial derivative, for the second difference - just the existence of partial derivative.

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