Compact-valued range doesnot imply compact graph

In summary, the conversation is discussing an example where y is a compacted valued correspondence of x, but the graph of x and y is not compact. Sin(x) is proposed as an example, but the speaker is unsure if it fits the criteria. The conversation also mentions the domains being compact and the possibility of finding a non-continuous function as an example.
  • #1
CHatUPenn
7
0
y is a correspondence of x. X is compact.
Can somebody give me an example where y is compacted valued, but the graph(x,y) is not compact.

A graph will be highly appreciated.
 
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  • #2
Maybe I'm misunderstanding your question but it seems y(x) = sin x has image [-1,1] yet graph(y) is not compact because unbounded..
 
  • #3
Quasar 987:
I just edited my question. Assuming X is compact,...
The statement is true even though the domain is compact.
I can tell you are doing physics. I am doing economics, sin function will never cross my mind.
Cheers
 
  • #4
Let me see if I understand what you're asking correctly. You want a function f:X->Y whose domain and range are compact, but whose graph isn't? If this is the case, then (unless you look at non-Hausdorff spaces) you won't be very lucky finding one that is continuous, so try to find that isn't continuous. (Another hint: Try a step function.)
 

1. What is a "compact-valued range"?

A compact-valued range is a set of values that is bounded and closed, meaning it contains all of its limit points. In other words, it is a range of values that is both finite and complete.

2. What does it mean for a compact-valued range to imply a compact graph?

If a function has a compact-valued range, it means that the output values of the function are bounded and closed. This does not necessarily mean that the graph of the function, which shows the relationship between the input and output values, will also be bounded and closed. Therefore, a compact-valued range does not always imply a compact graph.

3. Can you provide an example of a function with a compact-valued range but a non-compact graph?

Yes, consider the function f(x) = sin(x) on the interval [0, 2π]. The range of this function is the interval [-1,1], which is compact. However, the graph of this function is the entire sine curve, which is not bounded and closed.

4. Why is it important to understand the concept of compact-valued range and its relation to compact graphs?

Understanding the concept of compact-valued range and its relation to compact graphs is important in many areas of mathematics, including real analysis, topology, and functional analysis. It allows us to make precise statements about the behavior of functions and their graphs, and can help in proving theorems and solving problems.

5. How does compact-valued range relate to the concept of continuity?

A function with a compact-valued range is not necessarily continuous, and a continuous function does not necessarily have a compact-valued range. However, if a function is continuous and its domain is a compact set, then its range will also be a compact set. This is known as the intermediate value theorem, and it is one way in which compact-valued range and continuity are related.

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