Recall that for a function of one variable, the concept of differentiability at a point allowed us to approximate the function by a linear function in the neighborhood of . Analytically differentiable at is equivalent to the fact that
for all sufficiently small, where as . The expression
is the linear (or tangent line) approximation and is the error for the linear approximation. For function of two variables, the linear approximation motivates the following definition:
31.2.1
Definition:
A function
is said to be differentiable at if the following hold:
(i)
Both exist.
(ii)
There exists such that
(For example this condition will be satisfied if is an interior point of the domain )
of one variable, the concept of differentiability at a point 
allowed us to approximate the function 
by a linear function in the neighborhood of 
. Analytically 
differentiable at 
is equivalent to the fact that 
sufficiently small, where 
as 
. The expression 
is the error for the linear approximation. For function of two variables, the linear approximation motivates the following definition: 
if the following hold: 
exist. 
such that 
is an interior point of the domain 
) 
and 
such that 
