708 CHAPTER 22. THE DERIVATIVE

if and only iflimy→x

fk (y) = Lk (22.1.2)

where f(y)≡ ( f1 (y) , · · · , fp (y)) and L≡ (L1, · · · ,Lp).Suppose here that f has values in W, a normed linear space and

limy→x

f (y) = L, limy→x

g(y) = K

where K,L ∈W. Then if a, b ∈ F,

limy→x

(a f (y)+bg(y)) = aL+bK, (22.1.3)

If W is an inner product space,

limy→x

( f ,g)(y) = (L,K) (22.1.4)

If g is scalar valued with limy→x g(y) = K,

limy→x

f (y)g(y) = LK. (22.1.5)

Also, if h is a continuous function defined near L, then

limy→x

h◦ f (y) = h(L) . (22.1.6)

Suppose limy→x f (y) = L. If ∥ f (y)−b∥ ≤ r for all y sufficiently close to x, then |L−b| ≤ ralso.

Proof: Suppose 22.1.1. Then letting ε > 0 be given there exists δ > 0 such that if0 < ∥y−x∥< δ , it follows

| fk (y)−Lk| ≤ ∥f(y)−L∥< ε

which verifies 22.1.2.Now suppose 22.1.2 holds. Then letting ε > 0 be given, there exists δ k such that if

0 < ∥y−x∥< δ k, then| fk (y)−Lk|< ε.

Let 0 < δ < min(δ 1, · · · ,δ p). Then if 0 < ∥y−x∥< δ , it follows

∥f(y)−L∥∞< ε

Any other norm on Fm would work out the same way because the norms are all equivalent.Each of the remaining assertions follows immediately from the coordinate descriptions

of the various expressions and the first part. However, I will give a different argument forthese.