Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat, J. A. C. Kolk, J. P. van Braam Houckgeest

By J. J. Duistermaat, J. A. C. Kolk, J. P. van Braam Houckgeest

Quantity 1 offers a finished evaluation of differential research in multidimensional Euclidean area.

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3 below, we will verify that D f (a) is uniquely determined once it exists. We say that f is differentiable if it is differentiable at every point of its domain U . In that case the derivative or derived mapping D f of f is the operator-valued mapping defined by D f : U → Lin(Rn , R p ) with a → D f (a). ❍ In other words, in this definition we require the existence of a mapping R ⊃→ R p satisfying, for all h with a + h ∈ U , a : n f (a + h) = f (a) + D f (a)h + a (h) with lim h→0 a (h) h = 0. 10) 44 Chapter 2.

Let A ⊂ Rn and B ⊂ R p and let f : A → B be a bijection. Then the following are equivalent. (i) f is a homeomorphism. (ii) f is continuous and open. (iii) f is continuous and closed. Proof. (i) ⇔ (ii) follows from the definitions. ❏ At this stage the reader probably expects a theorem stating that, if U ⊂ Rn is open and V ⊂ Rn and if f : U → V is a homeomorphism, then V is open in Rn . 2. Moreover, Brouwer’s Theorem states that a continuous and injective mapping f : U → Rn defined on an open set U ⊂ Rn is an open mapping.

8. Compactness and uniform continuity 31 such that O does not contain a finite subcovering of K ∩ B1 = ∅. But this implies the existence of a rectangle B2 with B2 ∈ B2 , B2 ⊂ B1 , O contains no finite subcovering of K ∩ B2 = ∅. By induction over k we find a sequence of rectangles (Bk )k∈N such that Bk ∈ Bk , Bk ⊂ Bk−1 , O contains no finite subcovering of K ∩ Bk = ∅. 13) Now let xk ∈ K ∩ Bk be arbitrary. Since Bk ⊂ Bl for k > l and since diameter Bl = 2−l diameter B, we see that (xk )k∈N is a Cauchy sequence in Rn .

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