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Property of Monotonic Functions

  1. Jun 24, 2010 #1
    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 occured 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?
  2. jcsd
  3. Jun 24, 2010 #2


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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
  4. Jun 24, 2010 #3
    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]?
  5. Jun 24, 2010 #4
    That's right.
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