By George L. Ph.D., D.Sc. Clark

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This proposition is due to Kuratowski (see, for instance, [120]) and has numerous applications in topology and analysis. Lemma 1. , the mapping pr1 : X × Y → X is defined by the formula pr1 ((x, y)) = x ((x, y) ∈ X × Y ). , for each closed subset A of X × Y , the image pr1 (A) is closed in X. Proof. Take any point x ∈ X such that U (x) ∩ pr1 (A) = ∅ for all neighborhoods U (x) of x. We are going to show that x ∈ pr1 (A). For this purpose, it is sufficient to establish that ({x} × Y ) ∩ A = ∅. , ({x} × Y ) ∩ A = ∅.

This means that, for any point x ∈ G, there exists its neighborhood V (x) for which the set ∪{fi (V (x)) : i ∈ I} is bounded from above. Formulate and prove an analogous statement for upper semicontinuous functions. Notice that the result presented in Exercise 21 easily implies the wellknown Banach-Steinhaus theorem (see [14] or [81]). Exercise 22. Let E be a topological space, let f :E→R be a partial function bounded from above (respectively, from below) and suppose that dom(f ) = ∅. For any point x ∈ cl(dom(f )), let us put f ∗ (x) = limsupy→x,y∈dom(f ) f (y), and, respectively, f∗ (x) = liminfy→x,y∈dom(f ) f (y).

Let us demonstrate that, for each natural number n, the inclusion Z \ ∪{Di(k) : k ∈ N} ⊂ ∪{Di(k) : k ∈ N, k > n} © 2006 by Taylor & Francis Group, LLC 36 strange functions in real analysis holds true. Indeed, let z be an arbitrary point from Z \ ∪{Di(k) : k ∈ N}. Then, in particular, z ∈ Z \ (Di(0) ∪ ... ∪ Di(n) ). Because {Di : i ∈ I} is a Vitali covering of Z, there exists a segment Di for which z ∈ Di , Di ∩ (Di(0) ∪ ... ∪ Di(n) ) = ∅. Obviously, we have λ(Di ) > 0. At the same time, as mentioned above, the relation limk→+∞ λ(Di(k) ) = 0 is valid.