Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional derivatives
Lecture 31 :  Differentiability [Section 31.2]
 

and

31.2.2 Note(Necessary and Sufficient condition for differentiability):
 

Let be differentiable at Then by definition,

Hence,

                                 ----------(25)

In fact, the converse also holds, i.e., if (25) holds then is differentiable. We assume this fact.

As an application of the above equivalence, we have the following:

31 .2.3 Example:
(i)
 Let
 

Then is not differentiable at as both and do not exit.

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