Why doesn't this graph have a maximum value by the Extreme Value Theorem?

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I was provided this graph and asked if it passed the Extreme Value Theorem. I thought yes. I can see that this function is discontinuous...however, I was informed that this graph actually fails the Extreme Value Theorem due to the hole at x = 2. This caught me off guard, because I thought for certain there was a maximum.

Is the reason that there is no maximum for this graph because if someone were to say, "Well, the max is clearly 7.999"...another person could say, "Actually it's 7.9999"...and then another person could say...etc?

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2 Answers

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The maximum and minimum of a set are elements of the set. In your case the maximum is not attained and as you have explained any number close to $2$ is not a maximum.

That is why we have the notion of supremum of a set which is the least upper bound of the set and it does not have to belong to the set.

We say that $2$ is the supremum of the values of this function not a maximum.

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All the hypotheses of the EVT are not satisfied; in this case, your function is not continuous over the closed interval $[1,3]$.

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