Graphical Proof: The Max-Min Theorem Does Not Hold for Non-Continuous Functions

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In summary, the Max-Min Theorem is a mathematical theorem that states that for a continuous function on a closed interval, the function will have a maximum and minimum value on that interval. It is often used in graphical proofs to show the existence of a maximum or minimum value and cannot be applied to non-continuous functions. An example of a non-continuous function where the Max-Min Theorem does not hold is f(x) = 1/x on the interval [1, 3]. Other theorems, such as the Intermediate Value Theorem, can be used for non-continuous functions.
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Hi, just trying to do some homework for college and I can't get my head around this question. It is a question to show that if you take away the condition of a function being continuous, the max-min theorem no longer holds true. Any help is greatly appreciated!

Suppose that f: [a,b] -> R is not continuous. Show that f need not have an absolute maximum and an absolute minimum on [a,b]. (Answer in graphical form)
 
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Tell me about f(x)=1/(x-a) if x not equal to a and f(x)=0 if x=a. Then graph it.
 

1. What is the Max-Min Theorem?

The Max-Min Theorem is a mathematical theorem that states that for a continuous function on a closed interval, the function will have a maximum and minimum value on that interval.

2. How is the Max-Min Theorem used in graphical proofs?

The Max-Min Theorem is often used in graphical proofs to show the existence of a maximum or minimum value of a function on a given interval. It can also be used to help find these values.

3. What does it mean for the Max-Min Theorem to not hold for non-continuous functions?

This means that the Max-Min Theorem cannot be applied to non-continuous functions, as they do not have a maximum or minimum value on a closed interval. The function may have discontinuities, jumps, or other irregularities that prevent the theorem from being applied.

4. Can you provide an example of a non-continuous function where the Max-Min Theorem does not hold?

One example is the function f(x) = 1/x on the interval [1, 3]. This function is not continuous at x=0 and therefore does not have a maximum or minimum value on the interval, violating the Max-Min Theorem.

5. Are there any other theorems that can be used for non-continuous functions?

Yes, there are other theorems that can be used for non-continuous functions. One example is the Intermediate Value Theorem, which states that if a function is continuous on a closed interval, it takes on every value between its maximum and minimum values on that interval.

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