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# Differentiable

Let be differentiable on and continuous at . If

for then prove that is differentiable at .

Solution

Let us assume without loss of generality that . We will show that as and that means that is differentiable at with .

Fix and choose such that . Therefore . Now suppose . Let be the point that coincides with on the first coordinates and is elsewhere. Then is a path from to and each vector is parallel to one of the axes. Hence