Takeover Season
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I was just wondering, if we are calculating the infimum/supremum of a function's values, then if we find the 'maximum' or 'minimum' value of that function over its domain, can we always say this is equal to the supremum or infimum respectively?

Thank you. E.g. If we have a function with a domain X and a codomain of the real numbers, then if f(x) has its maximum value of f(x) = a, over its domain, then, all upper bounds u will such that be u >= a, so the least upper bound will be u = a?

Is this right, at least for continuous functions?
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mqb2766
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(Original post by Takeover Season)
I was just wondering, if we are calculating the infimum/supremum of a function's values, then if we find the 'maximum' or 'minimum' value of that function over its domain, can we always say this is equal to the supremum or infimum respectively?

Thank you. E.g. If we have a function with a domain X and a codomain of the real numbers, then if f(x) has its maximum value of f(x) = a, over its domain, then, all upper bounds u will such that be u >= a, so the least upper bound will be u = a?

Is this right, at least for continuous functions?
Reasonable discussion at
https://math.stackexchange.com/quest...the-difference
If the min/max exist, they are the inf/sup.
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RichE
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(Original post by Takeover Season)
I was just wondering, if we are calculating the infimum/supremum of a function's values, then if we find the 'maximum' or 'minimum' value of that function over its domain, can we always say this is equal to the supremum or infimum respectively?

Thank you. E.g. If we have a function with a domain X and a codomain of the real numbers, then if f(x) has its maximum value of f(x) = a, over its domain, then, all upper bounds u will such that be u >= a, so the least upper bound will be u = a?

Is this right, at least for continuous functions?
Just to add to mqb, a function can have a sup and no max, even a continuous one.
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mqb2766
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(Original post by RichE)
Just to add to mqb, a function can have a sup and no max, even a continuous one.
As well as an iinf but no min
exp(-x)
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Takeover Season
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Thanks for your comments, I understand the situation now. By definition of max/min of a set, they are equal to the sup/inf, if they exist. If the largest value of a function over its domain exists i.e. it is not tending towards a number, but is actually the number, then supremum and the max are equal to that number.
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