Proving that left limit for a function exists

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SUMMARY

The discussion centers on proving the existence of the left limit for a strictly increasing function f: (0,1) → R, under the condition that f(x) < M for all x in (0,1). The key conclusion is that the properties of strictly increasing functions ensure that as x approaches 1 from the left, the function f(x) converges to a limit, thus confirming the existence of the left limit, denoted as \lim_{x\to 1^{-}}f(x).

PREREQUISITES
  • Understanding of limits in calculus
  • Knowledge of strictly increasing functions
  • Familiarity with the notation of limits, specifically left-hand limits
  • Basic concepts of real analysis
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  • Study the properties of strictly increasing functions in detail
  • Learn about left-hand limits and their applications in real analysis
  • Explore the implications of bounded functions on limits
  • Investigate the concept of continuity in relation to increasing functions
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Mathematics students, particularly those studying calculus and real analysis, as well as educators looking to deepen their understanding of limits and function behavior.

Halen
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the question states:

suppose that f:(0,1)--->R is a (strictly) increasing function.
suppose also that there is a constant MeR such that f(x)<M for xe(0,1). prove that the left limit of f exists?

how would you use the information about the function being strictly increasing?

Thank you!
 
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Halen said:
the question states:

suppose that f:(0,1)--->R is a (strictly) increasing function.
suppose also that there is a constant MeR such that f(x)<M for xe(0,1). prove that the left limit of f exists?

how would you use the information about the function being strictly increasing?

Thank you!

When you say the left limit of f, do you mean:

[tex]\lim_{x\to 1^{-}}f(x)[/tex]

?
 
oh yes.. sorry!
 

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