Proving that left limit for a function exists

In summary, we are given that f:(0,1)-->R is a strictly increasing function and there is a constant M in R such that f(x)<M for all x in (0,1). We are asked to prove the existence of the left limit of f at x=1. The information about the function being strictly increasing can be used to show that the left limit exists by showing that the function approaches a finite value as x approaches 1 from the left.
  • #1
Halen
13
0
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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  • #2
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]

?
 
  • #3
oh yes.. sorry!
 

FAQ: Proving that left limit for a function exists

What is a left limit for a function?

A left limit for a function is the value that the function approaches as the input values approach from the left side. It is denoted by the notation lim f(x) as x approaches a- where "a-" indicates that the input values are approaching from the left side of a.

How do you prove that a left limit for a function exists?

To prove that a left limit for a function exists, you can use the definition of a left limit, which states that the limit exists if and only if the values of the function can be made arbitrarily close to the limit value by taking input values sufficiently close to the limit point from the left side. This can be shown using the epsilon-delta definition of a limit.

Can a function have a left limit but not a right limit?

Yes, a function can have a left limit but not a right limit. This occurs when the function approaches different values from the left and right sides of the limit point. In this case, the two-sided limit does not exist.

How does the existence of a left limit affect the continuity of a function?

The existence of a left limit does not necessarily guarantee the continuity of a function. A function is continuous at a point if and only if its left limit and right limit exist and are equal to the value of the function at that point. If the left limit exists but is not equal to the right limit, the function is not continuous at that point.

Are there any specific techniques for proving the existence of a left limit?

There are several techniques for proving the existence of a left limit, including the epsilon-delta definition of a limit, the squeeze theorem, and the use of one-sided limits. It is important to carefully analyze the function and its behavior at the limit point to determine the most suitable method for proving the existence of a left limit.

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