Applied math. Part 1: Integral by Bocconi

By Bocconi

Notes for Bocconi utilized Math essential half summarizing lecture notes and workouts.

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15(b). Thus, the assumption there that G is open is essential. We close this section with a circumspective remark. We did not use sequences in this section. Connectedness is one of the only properties of a metric space I know whose examination never uses the concept of a convergent sequence. 2. 2, that a subset E of R is an interval if and only if when a, b ∈ E and a < b, then c ∈ E whenever a < c < b? (3) If (X, d) is connected and f : X → R is a continuous function such that |f (x)| = 1 for all x in X, show that f must be constant.

9 (Urysohn’s4 Lemma). If A and B are two disjoint closed subsets of X, then there is a continuous function f : X → R having the following properties: (a) 0 ≤ f (x) ≤ 1 for all x in X: (b) f (x) = 0 for all x in A: (c) f (x) = 1 for all x in B. Proof. Define f : X → R by f (x) = dist (x, A) , dist (x, A) + dist (x, B) which is well defined since the denominator never vanishes. ) It is easy to check that f has the desired properties. 10. If F is a closed subset of X and G is an open set containing F , then there is a continuous function f : X → R such that 0 ≤ f (x) ≤ 1 for all x in X, f (x) = 1 when x ∈ F , and f (x) = 0 when x ∈ / G.

While he was there, World War I began, and he was unable to return to France. His health deteriorated further, depression ensued, and he spent the rest of his life on the shores of Lac L´ eman in Switzerland. It was there that he received the Chevalier de la Legion d’Honneur, and in 1922 he was elected to the Acad´ emie des Sciences. He published significant works on number theory and functions. He died in 1932 at Chamb´ ery near Geneva. Exercises 37 The details of this induction argument are left as Exercise 1.

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