a) Show that f is discontinuous at any x 6= 0.
b) Show that f is continuous at x = 0.
c) Show that f is differentiable at x = 0 and compute the value f 0 (0).
d) Show that f is not integrable on the interval [1, 2] (or any interval, but I don’t mind if you use that interval specifically).
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Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to f...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and conti...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
1 6. Where is the function f(x) { { - X4 if x # 0 discontinuous?...
1 6. Where is the function f(x) { { - X4 if x # 0 discontinuous? if x = 0 0 Is this a removable discontinuity? ex if x < 0 7. Where is the function f(x) discontinuous? x2 if x > 0 Is this a removable discontinuity? Is it a jump discontinuity? f(x) = {
III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that ...
analysis 2
III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6
III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6
Answer C 6. Let f be a continuous function on [0, oo) such that 0 f(z)...
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks)...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks)...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Determine the x-values, if any, at which the function is discontinuous. x2 -9 for x <...
Determine the x-values, if any, at which the function is discontinuous. x2 -9 for x < -1 h(x) = { 0 for -15xs1 x² +9 for x > 1 O A. - 1,1 OB. -1,0, 1 O c. 1 O D. None
Solve the problem 6 Hint- Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z)...
Solve the problem 6
Hint-
Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
1 6. Where is the function f(x) { { - X4 if x # 0 discontinuous? if x = 0 0 Is this a removable discontinuity? ex if x < 0 7. Where is the function f(x) discontinuous? x2 if x > 0 Is this a removable discontinuity? Is it a jump discontinuity? f(x) = {
analysis 2
III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6
III. Let f,g be Riemann integrable on [a, b). Show that, for any k>0, f (x)g(x)dr. IV. Show that eb 6
Answer C
6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Determine the x-values, if any, at which the function is discontinuous. x2 -9 for x < -1 h(x) = { 0 for -15xs1 x² +9 for x > 1 O A. - 1,1 OB. -1,0, 1 O c. 1 O D. None
Solve the problem 6
Hint-
Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...
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