An alternative definition is that a function f is differentiable at a point p in its domain if there exist functions fx and fy such that
limh --> 0 |f(p+h) - f(p) - fx(p)h1 - fy(p)h2|/|h| = 0
where h = (h1,h2). This condition can be written in terms of epsilons and deltas as well. We say that f is differentiable at p if for any ε > 0 there exists a &delta > 0 such that
|f(p+h) - f(p) - fx(p)h1 - fy(p)h2|/|h| < ε
-ε|h| + fx(p)h1 + fy(p)h2 < (f(p+h) - f(p)) < ε|h| + fx(p)h1 + fy(p)h2
whenever |h| < &delta. This definition of differentiabilty is similar in form to the definition of continuity. Recall that in the geometrical interpretation of continuity the challenge was to find a small enough &delta-disc domain at p such that the surface over the domain would lie between two horizontal plates a distance ε above and below f(p). In the geometrical interpretation of differentiability, as seen in the inequality above, the challenge is to find a &delta-disc domain, centered at p, that is small enough so that the surface over the domain lies between two cones, where the value of ε determines the slope of the cones.