Showing piece-wise function continuous

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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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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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