Question

Let f:[0,1]——>R be define by f(x)= x if x belong to rational number and 0 if...

Let f:[0,1]——>R be define by f(x)= x if x belong to rational number and 0 if x belong to irrational number and let g(x)=x

(a) prove that for all partitions P of [0,1],we have U(f,P)=U(g,P).what does mean about U(f) and U(g)?

(b)prove that U(g) greater than or equal 0.25

(c) prove that L(f)=0

(d) what does this tell us about the integrability of f ?

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

a. Let be any partition of [0,1]. Clearly

Which gives that U(f)=U(g)

b. We have for any partition P, now we take the partition 0<0.5<1,

then,

Hence

c. Let

Since irrationals are dense, we can find an irrational number in , Hence

which is true for all partition P. Hence L(f)=0

d. we have  , hence f is not integrable

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
28.8 Let f(x)=x^2 for x rational and f(x) = 0 for x irrational. (a) Prove f...
28.8 Let f(x)=x^2 for x rational and f(x) = 0 for x irrational. (a) Prove f is continuous at x = 0. (b) Prove f is discontinuous at all x not= 0. (c) Prove f is differentiable at x = 0.Warning: You cannot simply claim f '(x)=2x.
Let C [0,1] be the set of all continuous functions from [0,1] to R. For any...
Let C [0,1] be the set of all continuous functions from [0,1] to R. For any f,g ∈ C[0,1] define dsup(f,g) = maxxE[0,1] |f(x)−g(x)| and d1(f,g) = ∫10 |f(x)−g(x)| dx. a) Prove that for any n≥1, one can find n points in C[0,1] such that, in dsup metric, the distance between any two points is equal to 1. b) Can one find 100 points in C[0,1] such that, in d1 metric, the distance between any two points is equal to...
Show there does not exist a sequence of continuous functions fn : [0,1] → R converging...
Show there does not exist a sequence of continuous functions fn : [0,1] → R converging pointwise to the function f : [0,1] → R given by f(x) = 0 for x rational, f(x) = 1 for x irrational.
Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|. Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…). Prove that maps X onto...
Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|. Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…). Prove that maps X onto the Cantor set and satisfies (1/3)*d(x,y)≤|f(x)−f(y)|≤d(x,y) for x,y∈2N.
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0...
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show that there is a real number c such that f(x) = cg(x) for all x. (Hint: Look at f/g.) Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be the line tangent to the graph of g that passes through the point...
1. For n exists in R, we define the function f by f(x)=x^n, x exists in...
1. For n exists in R, we define the function f by f(x)=x^n, x exists in (0,1), and f(x):=0 otherwise. For what value of n is f integrable? 2. For m exists in R, we define the function g by g(x)=x^m, x exists in (1,infinite), and g(x):=0 otherwise. For what value of m is g integrable?
Let f(x) = x              0 ≤ x ≤ 1/2        = 3 - x         1/2 <...
Let f(x) = x              0 ≤ x ≤ 1/2        = 3 - x         1/2 < x ≤ 1 Find a partition P of [0,1] such that U(f, P) - L(f, P) < 1/100
Let G = {x in R | x > 0 and x notequalto 1}. Define *...
Let G = {x in R | x > 0 and x notequalto 1}. Define * on G by a * b = aln(b). Prove that the multiplicative group Rx is isomorphic to G.
(1) Let x be a rational number and y be an irrational. Prove that 2(y-x) is...
(1) Let x be a rational number and y be an irrational. Prove that 2(y-x) is irrational a) Briefly explain which proof method may be most appropriate to prove this statement. For example either contradiction, contraposition or direct proof b) State how to start the proof and then complete the proof
(b) Define f : R → R by f(x) := x 2 sin 1 x for...
(b) Define f : R → R by f(x) := x 2 sin 1 x for x 6= 0, and f(x) = 0 for x = 0. Does f 0 (0) exist? Prove your claim.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT