What Are Examples of Specific Sets and Functions in Analysis?

[dinhism]

Analysis Sets/Functions PLEASE PLEASE HELP! LIFESAVER if asnwered by 8am 12/22/09

Homework Statement

For each description below, provide a specific example fitting the description (provide some justification), or else explain why no such example exists.

1)An open set with no accumulation point

2)A subset of [0,\sqrt{2}] of Lebesgue measure 1 which contains no interval

3)A subset of [0,sqrt{2}] of Lebesgue measure 1 with no accumulation point

4)A bounded set with Lebesgue measure infinity

5)An open set with Lebesgue measure 0

6)A function that has all its derivative at p = 3 but is not analytic there

7)A sequence of functions on $R$, all continuous everywhere, all non differentiable at 0, that converge uniformly to a function differentiable everywhere

8)A series of functions which converges to (sin[3x])/x

Homework Equations

The Attempt at a Solution

I know that the answer to the first part is the null set, does anyone think that is wrong? please help me!

 

[HallsofIvy]

Homework Statement

For each description below, provide a specific example fitting the description (provide some justification), or else explain why no such example exists.

1)An open set with no accumulation point

2)A subset of [0,\sqrt{2}] of Lebesgue measure 1 which contains no interval

3)A subset of [0,sqrt{2}] of Lebesgue measure 1 with no accumulation point

4)A bounded set with Lebesgue measure infinity

5)An open set with Lebesgue measure 0

6)A function that has all its derivative at p = 3 but is not analytic there

7)A sequence of functions on $R$, all continuous everywhere, all non differentiable at 0, that converge uniformly to a function differentiable everywhere

8)A series of functions which converges to (sin[3x])/x

Homework Equations

The Attempt at a Solution

I know that the answer to the first part is the null set, does anyone think that is wrong? please help me!

Yes, the null set is an open set and it has no accumulation points. What more do you want?

If you honestly have no idea how to even begin any of the other problems, and don't even know the definitions of the important words, then you have a worse problem than we can help you with. Go to your teacher immediately!