By Avner Friedman
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Derivatives of Composite Functions j = 1, ... , n. (5) Since the components of 9 are differentiable at Xo by hypothesis, the first limit is the sum on the right in (I), and the proof can be completed by showing that the value of the second limit is zero. The element in the r-th row andj-th column of R is r[h; g(x o), g(xr)]lg(xr) - g(xo)l. In the denominator, apply the regularity condition satisfied by x. For r = I, ... , n, divide the denominator and the r-th row of R by IXr - Xo I. Since gl, ...
Elementary Properties of Differentiable Functions tion in all cases for j = 1, ... , n. xj(xo)](x j - x6). (48) Define r(j; xo, x n) as follows: r(j; x o , xo) (49) = O. 1] (51) Since Ix j - x61 ~ IXn- xol,j = 1, .. xj,j = 1, .. " n, are continuous at Xo by hypothesis. Then j= 1, "', n, (54) and each of the inequalities (52) and (53) shows that (51) is true. l9 is complete. 20 Corollary. Iff: E -+ IRm, E c IR n, I ~ m ~ n, is a function whose components (j1, .. ··,r) have partial derivatives in a neighborhood of Xo in E which are continuous at xo, thenfis differentiable at Xo with respect to every class X(xo, p), 0 < p ~ 1, and Du" ..
X n, x o) are in E. Let Xo be a point in E; then since E is open, there are n-vectors x: (Xl' ... , x n, x o) in E in every neighborhood N(x o, e) of xo. If x is in E, the functionfhas increments at Xo which correspond to the increment ~(x) of x at Xo' The following examples illustrate the general definition which follows. 5 Examples. Let x: (XI' ... , x n, x o) be an n-vector in E in IRn. If m = 1 and n = 3, then f has a single component (denoted by f); there are three 18 I. Differentiable Functions and Their Derivatives increments of fat Xo which correspond to the increment Ll(x) of x at x o , and which are denoted by Lld(x), Ll 2f(x), Ll 3f(x) and defined as follows [compare (21) in Section 1].