Do Right and Left Hand Limits Exist for All Increasing Functions?

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Homework Help Overview

The discussion revolves around the existence of right and left hand limits for increasing functions, specifically questioning whether these limits must exist at every point in the real numbers. The original poster is attempting to prove this concept and has encountered difficulties in their reasoning.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster considers a proof by contradiction but struggles to find a viable path. They also mention the challenge of proving the statement directly. Other participants inquire about the definitions of left and right hand limits as provided in the original text, and the implications of the increasing nature of the function on these limits.

Discussion Status

Participants are exploring the definitions of limits in the context of increasing functions and questioning how these definitions relate to the existence of limits. There is an acknowledgment of confusion regarding the application of these definitions, and the original poster expresses a desire to revisit the problem after further consideration.

Contextual Notes

The original poster references a specific theorem from a PDF document that may provide additional context or support for their inquiry. There is an implication that the definitions of limits are critical to the discussion, and the increasing property of the function is a central assumption being examined.

SomeRandomGuy
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Been bangin my head against the wall all weekend thinkin about this question:

Let f:R->R be an increasing function. Proce that lim x->c+f(x) and lim x-c-f(x) (right and left hand limits) must each exist at every point c in R.

There's more to the question, but if I can get this part solved, I'm sure the rest won't be trouble.

My original idea was to prove this by contradiction, assuming the limits don't exist, and showing this violates the increasing aspect. However, I've come to deadends each time. Proving it directly seems very difficult as well.
 
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Has your text defined left/right-handed limits of f at x in terms of what happens to f\left( x_n\right) for alll sequences that converge to x which are strictly less than/greater than x?
 
benorin said:
Has your text defined left/right-handed limits of f at x in terms of what happens to f\left( x_n\right) for alll sequences that converge to x which are strictly less than/greater than x?

Yes,

limx->c+ f(x) = L if limf(x_n)=L for all x_n > c where lim(x_n)=c.

Similar definition for for limx->c- f(x).

Since f was defined as an increasing function, then x_n is either increasing or decreasing depending on what limit we are taking. How can we conclude that limf(x_n) = L? Still kind of confused. I'll go work on it and check back in the morning.

Thanks for the help.
 

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