![5. Let the functions fon : [a, b] → R be uniformly bounded continuous func- tions. Set di, astsb. Prove that F, has a uniform](http://img.homeworklib.com/questions/03dcf440-6ad0-11eb-a169-53d04f4ca9ca.png?x-oss-process=image/resize,w_560)
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![Solution - Giver function fr : [a,b]-DR It is uniformly bounded continuous function And also Given set x FA (2)= faltat agxsb](http://img.homeworklib.com/questions/2847bf60-6ad0-11eb-81cf-7f92bfe0f8d6.png?x-oss-process=image/resize,w_560)
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5. Let the functions fon : [a, b] → R be uniformly bounded continuous func- tions....
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
Number 6 please S. Let ) be a sequence of continuous real-valued functions that converges uniformly...
Number 6 please
S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
2.) (b): Prove or disprove the following problems. 1. Suppose fn(x) is uniformly convergent to fon...
2.)
(b): Prove or disprove the following problems. 1. Suppose fn(x) is uniformly convergent to fon D= [a, b]. Let ce [a, b]. Is fr uniformly convergent to f on D1 = (a, and/or D2 = (c, b)? = (a, and D2 = [c, b). Is in 2. Let a <c<b. Suppose fn(x) is uniformly convergent to f on D uniformly convergent to f on D = (a,b). 3. Suppose that fn(a) is uniformly convergent to fon , i=1,2,... Is...
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R...
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R be continuous. (c) If h is uniformly continuous on R, then goh is uniformly continuous. (b) If g(x) = {g(x) : x € X} is a bounded set,(i.e. there exists M > 0 such that g(x) < M for all X E X.) then go h is uniformly continuous.
Mathematical Analysis help 2. If J is uniformly continuous on a bounded subset A of R,...
Mathematical Analysis help
2. If J is uniformly continuous on a bounded subset A of R, prove that f is bounded on A.
2. Let f:R + R and g: R + R be functions both continuous at a...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for ...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 fo...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
2. Let SCR be a measurable regular region, and let L(S) be the vector space of bounded continuous...
2. Let SCR be a measurable regular region, and let L(S) be the vector space of bounded continuous functions f : S-> R satisfying el Prove that |is a nor on L(R)
2. Let SCR be a measurable regular region, and let L(S) be the vector space of bounded continuous functions f : S-> R satisfying el Prove that |is a nor on L(R)
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
Number 6 please
S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
2.)
(b): Prove or disprove the following problems. 1. Suppose fn(x) is uniformly convergent to fon D= [a, b]. Let ce [a, b]. Is fr uniformly convergent to f on D1 = (a, and/or D2 = (c, b)? = (a, and D2 = [c, b). Is in 2. Let a <c<b. Suppose fn(x) is uniformly convergent to f on D uniformly convergent to f on D = (a,b). 3. Suppose that fn(a) is uniformly convergent to fon , i=1,2,... Is...
Let X be metric space, and let g:X + R be uniformly continuous and h: R+R be continuous. (c) If h is uniformly continuous on R, then goh is uniformly continuous. (b) If g(x) = {g(x) : x € X} is a bounded set,(i.e. there exists M > 0 such that g(x) < M for all X E X.) then go h is uniformly continuous.
Mathematical Analysis help
2. If J is uniformly continuous on a bounded subset A of R, prove that f is bounded on A.
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
2. Let SCR be a measurable regular region, and let L(S) be the vector space of bounded continuous functions f : S-> R satisfying el Prove that |is a nor on L(R)
2. Let SCR be a measurable regular region, and let L(S) be the vector space of bounded continuous functions f : S-> R satisfying el Prove that |is a nor on L(R)
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