Locally bounded argument

In summary, the conversation discussed the function f(x) = 1/x being locally bounded at each point x in the set E = (0,1). The argument presented was that this function is false since the point 0 is an accumulation point that is not bounded. However, this can be avoided by making E closed. The last part of the conversation questioned the significance of 0 not being bounded and whether the statement locally bounded implies bounded is a counterexample.
  • #1
ocohen
24
0
Having a hard time understanding this example from a book:

The function f(x) = 1/x is locally bounded at each point x in the set E = (0,1).
Let x \in (0,1). Take \delta_x = x/2, M_x = 2/x. Then
f(t) = 1/t <= 2/x = M_x
if
x/2 = x-\delta_x < t < x + \delta_x

This argument is false since the point 0 is an accumulation point that does not belong to (0,1). As such there is no assumption that 0 is bounded. This can be avoided by making E closed.

I don't understand this last part. Why does it matter that 0 is not bounded? I thought the whole point of locally bounded is that we can define the bound in terms of x. And we can always find a small enough interval (namely x/2) that will not include 0.

Can anyone shed some light on this?

Thanks
 
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  • #2
It sounds like this is supposed to be a counterexample to the statement locally bounded implies bounded. On the other hand if the domain is compact and the function is locally bounded and continuous, then the function is bounded

EDIT: In retrospect that's not very enlightening (since the locally bounded part is completely unnecessary). I'm not sure what the point is
 

1. What is a locally bounded argument?

A locally bounded argument is a type of logical argument in which the premises and conclusion are all statements about specific, finite portions of a larger set or domain. This means that the argument is only valid within a specific context or boundary, and may not hold true outside of that context.

2. How is a locally bounded argument different from a globally bounded argument?

A globally bounded argument is a type of logical argument in which the premises and conclusion are statements about the entire set or domain. This means that the argument is valid in all contexts or boundaries, and holds true universally. In contrast, a locally bounded argument is only valid within a specific context or boundary.

3. What are some examples of locally bounded arguments?

Examples of locally bounded arguments include mathematical proofs that are only valid within a specific set of numbers or equations, scientific arguments that are only applicable to a certain species or environment, and legal arguments that are only relevant to a specific jurisdiction or set of laws.

4. How are locally bounded arguments used in scientific research?

Locally bounded arguments are commonly used in scientific research to make conclusions about specific observations or experiments within a controlled context. These arguments may then be used to support or refute broader theories or hypotheses, but are limited in their generalizability outside of the specific context in which they were tested.

5. What are the potential limitations of using locally bounded arguments?

One limitation of locally bounded arguments is that they may not accurately represent the larger, global truth. By focusing on a specific context or boundary, these arguments may overlook important factors that may impact the validity of the conclusion in a larger context. Additionally, the validity of a locally bounded argument may be limited by the quality and accuracy of the data or observations within the specific context.

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