Limits and Continuity - Absolute Value Technicality ....

In summary, the absolute value technicality is a method used to determine the continuity of a function at a specific point. It involves evaluating the limit on both sides of the point and checking if they are equal. However, this method can only be used for functions that are piecewise continuous and may not be applicable in cases of jump discontinuities. Other methods, such as the epsilon-delta definition and the intermediate value theorem, are used to evaluate continuity over an interval or on a whole function.
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I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ...

I need some help with an inequality involving absolute values in Example 4.1.2 (a) ...

Example 4.1.2 (a) ... reads as follows:

?temp_hash=3b7c8f68814a2778b219da7337e2fc04.png
In the above text we read ...

"... If ## \mid x - p \lvert \ \lt 1## then ##\mid x \mid \lt \mid p \mid + 1 ## ... "Can someone please show me how to rigorously prove the above statement ...

Peter
 

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Just use the triangle inequality.
 
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##|x| = |x-p+p| \leq |x-p| + |p| \Rightarrow |x| - |p| \leq |x-p| < 1 \Rightarrow |x| < 1 + |p|##
 
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Related to Limits and Continuity - Absolute Value Technicality ....

1. What is the absolute value technicality in limits and continuity?

The absolute value technicality in limits and continuity refers to the behavior of a function when approaching a specific point or value. It is used to determine if a function is continuous at a certain point or if it has a discontinuity.

2. How is the absolute value technicality used in determining continuity?

The absolute value technicality is used to check for continuity at a specific point by evaluating the limit on both sides of the point. If the limits from both sides are equal, then the function is continuous at that point. However, if the limits from both sides are not equal, then the function has a discontinuity at that point.

3. Can the absolute value technicality be used to determine the continuity of all functions?

No, the absolute value technicality can only be used to determine the continuity of functions that are piecewise continuous. This means that the function must have a defined value at the specific point being evaluated.

4. How does the absolute value technicality differ from other methods of evaluating continuity?

The absolute value technicality is a specific method used to evaluate continuity at a specific point, whereas other methods such as the epsilon-delta definition or the intermediate value theorem are used to determine continuity over an interval or on a whole function.

5. Are there any limitations to using the absolute value technicality in determining continuity?

Yes, the absolute value technicality may not be applicable in cases where the function is not piecewise continuous or has a jump discontinuity. In these cases, other methods of evaluating continuity may need to be used.

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