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Wolfram defines Global Maxima as :

A global maximum, also known as an absolute maximum, the largest overall value of a set, function, etc., over its entire range.

As per the definition, I'm not sure if $\infty$ can be considered as the global maxima for functions like $x$, $x^2$, etc. What if we consider the extended real number set?

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    $\begingroup$ No. Conventionally "maximum" always means a finite number. In extended real number system though, we accept $\infty$ as possible value for "supremum", but still a "maximum" is always finite. $\endgroup$ – Vim Jul 1 '18 at 8:40
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    $\begingroup$ @Vim Exactly. The consideration of infinity as supremum led me to such a doubt. $\endgroup$ – Mathejunior Jul 1 '18 at 8:41
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    $\begingroup$ @Vim: That's an answer, not a comment. $\endgroup$ – Hans Lundmark Jul 1 '18 at 9:07
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You've probably found that when you write infinity in interval notation, you are instructed to write for example $$ y\in [0,\infty) $$ Note how the high bound infinity is not included. This is because infinity is a boundary not a real number, $$\infty \not\in \mathbb{R}$$ that is a number can approach but not touch infinity, so it could not be that the maximum is infinity.

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