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

[wayneckm]

Hello all,

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

Am I correct on the above statement?

 

[Office_Shredder]

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 $$f(x) \leq f(y)$$ or $$f(x) \geq f(y)$$.

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

 

[wayneckm]

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 $$f$$ on $$\mathbb{R}$$?

 

[Tedjn]

That's right.