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First let assume we got the following function: $$f:{\bf R}^{n}\longrightarrow {\bf R}^{m}$$ Leaving the formal definition of differentiability at a point away, I trying to find out more practical conditions for differentiability at a point of multivariable function.

One of those practical theorems is that if the partial derivatives of the the function are continuous at all points, then the function is differentiable.

Now, I trying to understand if there are conditions for differentiability at a particular point (assume the point $\vec a$), in more precise, my question is that if it's true to say that if the partial derivatives are continuous in small neighborhood of $\vec a$ then the function is differentiable at the point?

if it is not true: does it matter if $m>1$ or not? And moreover, does "practical" theorem of conditions differentiability at a point is exist?

I was trying to find out whether this statement is true or not, without any success.

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The statement is true ! The corresponding theorem and a proof can be found in

Mathematical Analysis, by Tom M. Apostol, Theorem 6-18.

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