Showing piece-wise function continuous

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Homework Help Overview

The discussion revolves around the continuity of a piecewise function, specifically examining the function defined as \( f(x) = \cos x \) on the interval \([\frac{\pi}{4}, \infty)\) and its behavior around the point \(\frac{\pi}{4}\).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore whether certain details, such as explicitly stating the function on specific intervals, are necessary for clarity. There is also a discussion about the importance of including or omitting certain points when discussing continuity.

Discussion Status

The conversation is ongoing, with participants providing feedback on the clarity of statements made in the context of the function's continuity. Some guidance has been offered regarding the inclusion of specific points in the discussion.

Contextual Notes

There is a focus on the continuity of the function across specified intervals, with particular attention to the implications of including or excluding the point \(\pi/4\) in the analysis.

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Homework Statement
Please see below
Relevant Equations
Please see below
For this,
1680662250322.png
,
The solution is,
1680662269391.png

However, should they not write ##f(x) = \cos x## on ##[\frac{pi}{4}, \infty)##

Many thanks!
 
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Do you mean right after the "Similarly,"? It wouldn't hurt, but I think that it is easy enough to follow the logic without saying that. Initially, you should be in the habit of stating everything. After a while, that becomes tedious and both you and the reader will be happy if you skip obvious things. You must be careful though, what you skip.
 
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FactChecker said:
Do you mean right after the "Similarly,"? It wouldn't hurt, but I think that it is easy enough to follow the logic without saying that. Initially, you should be in the habit of stating everything. After a while, that becomes tedious and both you and the reader will be happy if you skip obvious things. You must be careful though, what you skip.
Thank you for you reply @FactChecker!

No sorry I meant right after the "Since f(x) = Sinx on ..."

Many thanks!
 
ChiralSuperfields said:
Thank you for you reply @FactChecker!

No sorry I meant right after the "Since f(x) = Sinx on ..."

Many thanks!
Oh. The reason for not including the point ##\pi/4## is that the rest of the sentence is only about continuity on ##(-\infty, \pi/4)\cup(\pi/4, \infty)##. So there was no need to include ##\pi/4##. It wouldn't have hurt to include it.
 
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