Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional derivatives
Lecture 31 :  Partial derivatives  [Section 31.1]
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 Let
 

Then

and at ,

Thus, both the partial derivatives of exist everywhere. However, is not continuous at . To see this, note that along the line ,

Hence

31.1.4 Note:
 

Examples 31.1.2 show that the existence of both the partial derivatives at a point need not imply continuity of the function at that point. The reason being that the partial derivatives only exhibit the rate of change of only

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