What Are Examples of Discontinuous Functions and Their Properties?

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The discussion centers on examples of discontinuous functions, specifically a piecewise function f(x)=1/x that is discontinuous on the interval [-4,4] and does not take the value c=0. Participants also explore the concept of nested sequences of non-compact sets with an empty intersection and an unbounded infinite set lacking accumulation points. One user proposes the set {1/n} U {0} to eliminate accumulation points, seeking clarification on its validity. The thread emphasizes the importance of understanding the underlying mathematical concepts rather than simply obtaining answers.
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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 o:) )
 
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https://www.physicsforums.com/showthread.php?t=4825
 
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.
 
I'm just here for help. If you don't want to help, then please don't reply.
 

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