The condition that exists near and are continuous at are only sufficient for the differentiability of at . These are not necessary. For example, the function
is differentiable at , as shown in example . However,
which does not converge to as . In fact, along the path the function is unbounded.
Practice Exercises
(1)
Examine the following functions for differentiability at . The expressions below give the value of the
function at . At , the value should be taken as zero.
(i)
(ii)
exists near 
and are continuous at 
are only sufficient for the differentiability of 
at 
. These are not necessary. For example, the function 
, as shown in example 
. However, 
as 
. In fact, along the path 
the function 
is unbounded. 
. The expressions below give the value of the
. At 
, the value should be taken as zero. 
