(Expand this Subsection to read an explanation of the technical definition)

Here again, for your reading pleasure, is the  technical definition of differentiability: a function is differentiable at a point a if there's a linear transformation T such that

 Underscript[lim, x  a] (|| f (x) - f (a) - T (x - a) ||)/(|| x - a ||) = 0

(The double bars in that equation mean ``length of a vector'.')  Let's think about what that equation means: the limit is zero, so as x approaches a, the numerator becomes much smaller than the denominator. The denominator is just the distance from x to a. Since T is linear, T(x-a) = T(x) - T(a), and the part of the numerator inside the bars can be written in a somewhat more illuminating way:

f (x) - f (a) - (T (x) - T (a)) = f (x) - f (a) - T (x) + T (a) .

But wait...T is a tangent plane (or ``tangent 3-space'', etc) so f(a) must equal T(a), and in the above equation they'll cancel. So the numerator becomes even simpler—

f (x) - T (x)

—and we can rewrite the above limit as

RowBox[{ , RowBox[{Underscript[lim, x  a] (| f (x)    - T (x) |)/(| x - a |),  , =,  , 0.}]}]

If that limit is zero, then as x approaches a (along any path, of course) the numerator is much smaller than the denominator, which means that the linear transformation approximates f(x) very well near a. You might say that in a very small neighborhood of a, f(x) is ``just about'' linear. This is what we were trying to visualize earlier in this section.

This definition of differentiability is, unfortunately, not very practical. If we're interested in deciding whether a function is differentiable or not (and also in finding the particular linear transformation that approximates it), we have to use a test which will be described below.


Created by Mathematica  (November 6, 2004)