Is the Supremum/Infimum of Monotonic Functions Always at Infinity?

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Discussion Overview

The discussion revolves around the behavior of supremum and infimum for monotonic functions, specifically whether these values are always found at infinity. The scope includes theoretical aspects of monotonicity and its implications for maximum and minimum values.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that for a monotonic increasing or decreasing function on the real numbers, the supremum or infimum occurs at infinity, as a maximum or minimum would contradict monotonicity.
  • Another participant introduces the distinction between strict and weak monotonicity, explaining that under weak monotonicity, a function can achieve a maximum or minimum if it becomes constant after a certain point.
  • A later reply clarifies that the initial assertion was intended to refer to strictly monotonic functions, suggesting that such functions do not have maximum or minimum values on the real line.
  • It is noted that the limit as x approaches infinity does not necessarily exist, indicating that a function may be unbounded.

Areas of Agreement / Disagreement

Participants generally agree that strictly monotonic functions do not have maximum or minimum values. However, there is a recognition of differing definitions of monotonicity, leading to multiple views on the behavior of weakly monotonic functions.

Contextual Notes

The discussion highlights the dependence on definitions of monotonicity and the conditions under which limits exist, which remain unresolved.

wayneckm
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Hello all,

For a monotonic increasing/decreasing function [tex]f(x)[/tex] on [tex]x \in \mathbb{R}[/tex], we can only have supremum/infimum which is occurred at [tex]x = \infty[/tex] with value [tex]\lim_{x\uparrow \infty}f(x)[/tex] Otherwise, if it was a maximum/minimum, it would violate the assumption of monotonicity.

Am I correct on the above statement?
 
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It depends on your definition of monotonic. There are notions of strict and weak monotonicity; strict means that the function is strictly increasing/decreasing, i.e. x<y means f(x)<f(y) or f(x)>f(y).

Weak means only that x<y means [tex]f(x) \leq f(y)[/tex] or [tex]f(x) \geq f(y)[/tex].

If you're only looking at weak monotonicity, then you can have the function be constant after some value and achieve a maximum/minimum value. For strong monotonicity this can't occur

Also note that the limit as x goes to infinity doesn't have to exist (which means that the function is unbounded).

Obviously the same stuff applies as x goes to minus infinity also for the other bound
 
Thanks for the reply.

Ya, I forgot to state in my sense it is in strict mode.

So in summary, there does not exist maximum/minimum for a strictly increasing/decreasing function in the case of function [tex]f[/tex] on [tex]\mathbb{R}[/tex]?
 
That's right.
 

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