By B. Sendov (auth.), Gerald Beer (eds.)

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19. Let f, g e An. 40) cp(x) - c ~ ~ ",(x) g(x) ~ ",(x) +c hold. 41) r(a, a; f, g) ~ r(a, a; cp, 'II) + c. Proof. 4. 39) and - cp(x) + c) ~ f(x) - g(~) ~ ",(x) - cp(~) + c. Relationsbips between uniform distance and tbe Hausdorff distance The uniform distance in the set Cn of all continuous functions on follows: a is defined as R(f, g) =IIf - glln = sU'Q.. If(x) - g(x)1 xell It is possible to extend this definition to all the functions in expression su'Q.. 42) sU'Q.. 43) of course agrees with the usual definition.

The modulus of H-continuity with weight 'C(O, a, f; ° is given by a) = r(0, a; S(O, &2, t), 1(0, an, t). 60) that lim a-++ 0 t(O, a, f; a) = co(O, f; a). e. All statements for H-distance hold true for H-distance with weight as well, with uniform distance replaced by uniform distance with weight. Thus. 3. 10. If f. e. r(0, a; f, g) is the Hausdorff distance with weight between f and g. and R(O; f. g) is the uniform distance with weight between f and g. and n is compact, then CHAPTER 2 48 r(9, a; f, g) S R(9; f, g) S r(9, a; f, g) + ro(9; ar(9, a; f, g), where ro(9, 8) =min (ro(9, f; 8), ro(9, g; 8)}.

Let A, B c: M. If for a certain point a e A we have U(a, a) 0, then r(p; A, B) ~ a. nB = Proof. From the defInition of U(a, a) and the hypotheses of the lemma, it follows that for every point be B we have p(a, b) ~ a. As a result, inf p(a, b) ~ 0 so that beB sup. inf p(a, b) ~ a. 5), the assertion of the lemma follows. 2. Let A, B c: M, satisfying the following conditions: 25 HAUSDORFF DISTANCE (a) for every point a e A there exist points a' e M and b e B such that a, b e U(a, a'); (b) for every point b e B there exist points b' e M and a e A such that a, b e U(a, b').

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