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

Let

 

Then and . Hence,

Also

Hence, is differentiable at

As in the case of function of one variable, above notion of differentiability implies continuity of the function.

31.2.4 Theorem:
 

Let be such that is differentiable at . Then is continuous at .

 
31.2.5

Increment Theorem (Sufficient condition for differentiability):

  Let and be such that the following hold:
(i)

For some both and exist at all points in

(ii)
 Both and are continuous at the point . Then, is differentiable at .
 
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