Lets say f(x) is a one-variable function. Now let P=(c,0) and Q=(d,0) be points on the x-axis, such that f is defined on both P and Q. Let
Of course any approximation function also has its error function, which in this case is equal, for any point Q to,
Let's formally define a linear approximation.
Definition: Let f(x) be a one-variable function on a domain D. Let x=c be in D and let P, Q the change in x be defined as above. We say f has a linear approximation at c if, for our chosen c, there is a constant A and an error function of Q, such that:
This definition implies
The definition for linear approximation for a multi-variable function will be (not surprisingly) similar to that of one variable.
Definition: Let f( x_{1},x_{2},...x_{n}) be a n-variable function on domain D. Let P be a fixed point in D,
P=(a_{1},a_{2},...a_{n}).
Let Q=((a_{1}+x_{1}),(a_{2}+x_{2}),...(a_{n}+x_{n})) be another point in D.
Let f = f((a_{1}+x_{1}),(a_{2}+x_{2}),...(a_{n}+x_{n}))-f(a_{1},a_{2},...a_{n}).
We say that f has a linear approximation at P if, for our chosen P, there exist constants
A_{1},A_{2},...A_{n} and
n functions erf_{1}(Q), erf_{2}(Q),... erf_{n}(Q) such that
This definition implies the following:
One must note that it is possible for f to have first-order partial derivatives at every point, but not to have linear approximations at a certain point P in its domain. That is, a f can have first-order partial derivatives at every point, but not be continuous at a certain point P in its domain.