What Are Examples of Discontinuous Functions and Their Properties?

[Kraziethuy]

1. An example of a discontinuous function on an interval [a,b] that does not assume every value between f(a) and f(b).

[edit] My answer to this: Piecewise function f(x)= 1/x, for x greater than and equal to -4 but less than zero (0). And f(x)=1/x for x greater than zero but less than and equal to 4. This makes the function discontinuous, on the interval [-4,4]. Now, I let c=0. There does not exist a value x in (-4,4) such that f(x)=c. Correct?

2. Find a nested sequence of non-compact sets whose intersection is empty.

3. An example of an unbounded infinite set that has no accumulation point.

For this one, I know that 1/n, for n=1,2,3,... has ONLY zero as an accumulation point, so can I maybe do something like {1/n}U{0} so that there is no longer an accumulation point? I'm pretty suck on this one.

[edit] (I figured out #4 and #5 now )

 

[vincentchan]

we won't do HW for you, read the sticky

https://www.physicsforums.com/showthread.php?t=4825

 

[Kraziethuy]

My other thread went unanswered for the most part, and I'd rather not explain this one if it's going to end up the same way. If I made it this far in math, it's obviously not because I don't do my own hw.

I've spent two days already working with these problems and the other two listed in my previous thread. I know the basic theorems used in deciphering the problems, but haven't built up anything to go along with what I've posted.

I know it looks as though I'm just going for a quick answer, but even then I'd have to prove the answer received anyway. So there aren't free answers here.

 

[vincentchan]

https://www.physicsforums.com/showthread.php?t=59240
I have encounter a person like you a couple days ago... and warned my some staff in this forum...

 

[Kraziethuy]

I'm just here for help. If you don't want to help, then please don't reply.