The relationship between con't function and a compact set

In summary, the conversation discusses whether a map f:R^m -> R^n is continuous if its preimage set is compact for any compact set K in R^n. The answer is no, as shown by a counterexample function f:R->R. The function is defined as f(x)=log|x| if x is not equal to 0 and f(x)=0 if x=0. It is mentioned that the image of log|x| is difficult to visualize, and the speaker plans to try plugging in numbers to understand it better.
  • #1
pantin
20
0
suppose f:R^m -> R^n is a map such that for any compact set K in R^n, the preimage set f^(-1) (K)={x in R^m: f(x) in K} is compact, is f necessary continuous? justify.

The answer is no.
given a counterexample,

function f:R->R

f(x):= log/x/ if x is not equal to 0
f(x):= 0 if x=0

note, /x/ is the absolute value of x.


I don't quite get how to draw the image log/x/
and anyone can explain why ?
 
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  • #2
What exactly are you having trouble with?
 
  • #3
let me try to plug in some number to the fn in the solution tomorrow...too late tonight, going to sleep.. thanks for asking :)
 

1. What is a continuous function?

A continuous function is a mathematical function that has no abrupt changes or breaks in its graph. This means that as the input values of the function change, the output values change smoothly and continuously.

2. What is a compact set?

A compact set is a subset of a mathematical space that is closed and bounded. This means that the set contains all of its limit points and can be contained within a finite region.

3. How are continuous functions and compact sets related?

The relationship between continuous functions and compact sets is that a continuous function on a compact set must have a maximum and minimum value. This is known as the extreme value theorem.

4. What is the importance of understanding the relationship between continuous functions and compact sets?

Understanding this relationship is important in many areas of mathematics and science, as it allows for the prediction and analysis of real-world systems. It also provides a foundation for more complex mathematical concepts and theories.

5. Can a continuous function on a non-compact set have a maximum or minimum value?

No, a continuous function on a non-compact set does not necessarily have a maximum or minimum value. This is because the non-compact set does not have boundaries that the function must adhere to, allowing it to potentially increase or decrease without limit.

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