Page 164 - Vector Analysis

160 CHAPTER A. Appendix

The Completeness Axiom（實數完備性公設）

Every subset of R which is bounded from above has a least upper bound.

Definition A.6 (Cauchy sequence). A sequence txku8k=1 in R is said to be Cauchy if for
every ε ą 0, there exists N ą 0 such that |xk ´ xℓ| ă ε whenever k, ℓ ě N .
Theorem A.7. Every Cauchy sequence in R converges.

A.2 Properties of Continuous Functions

Theorem A.8 (Uniform Continuity).

Theorem A.9 (Mean Value Theorem).

Theorem A.10 (Inverse Function Theorem). Let f : (a, b) Ñ R be differentiable, and f 1

is sign-definite; that is, f 1(x) ą 0 for all x P (a, b) or f 1(x) ă 0 for all x P (a, b). Then

f : (a, b) Ñ f ((a, b)) is a bijection, and f ´1, the inverse function of f , is differentiable on

f ((a, b)), and

(f ´1)1(f (x)) = 1 @ x P (a, b) . (A.1)
f 1(x)

Proof. W.L.O.G. we assume that f 1(x) ą 0 for all x P (a, b). Then f is strictly increasing;

thus f ´1 exists.

Claim: f ´1 : f ((a, b)) Ñ (a, b) is continuous.

Proof of claim: Let y)0 = f (x0) P f ((a, b)), and εą0 be given. Then f ((x0 ´ ε, x0 + ε)) =
(
f (x0 ´ ε), f (x0 + ε) since f is continuous on (a, b) and (x0 ´ ε, x0 + ε) is connected. Let

δ = mintf (x0) ´ f (x0 ´ ε), f (x0 + ε) ´ f (x0)(. Then δ ą 0, and

()
(y0 ´ δ, y0 + δ) = f (x0) ´ δ, f (x0) + δ Ď f ((x0 ´ ε, x0 + ε)) ;

thus by the injectivity of f ,

f ´1((y0 ´ δ, y0 + δ)) Ď f ´1(f ((x0 ´ ε, x0 + ε))) = (x0 ´ ε, x0 + ε) = (f ´1(y0) ´ ε, f ´1(y0) + ε) .

The inclusion above implies that f ´1 is continuous at y0.
Writing y = f (x) and x = f ´1(y). Then if y0 = f (x0) P f ((a, b)),

f ´1(y) ´ f ´1(y0) = x ´ x0 .
y ´ y0 f (x) ´ f (x0)