Showing piece-wise function continuous

In summary, the conversation is discussing whether or not to include the point pi/4 in a solution, with the speaker stating that it is not necessary since the rest of the sentence is only about continuity on (-infinity, pi/4)U(pi/4, infinity).
  • #1
ChiralSuperfields
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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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  • #2
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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  • #3
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!
 
  • #4
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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1. What is a piece-wise function?

A piece-wise function is a mathematical function that is defined by different equations over different intervals. In other words, the function is divided into different pieces, and each piece is described by a different equation.

2. How do you show a piece-wise function is continuous?

To show that a piece-wise function is continuous, you must first check that each piece of the function is continuous within its respective interval. Then, you must check that the function's value at the endpoint of each interval is equal to the limit of the function as it approaches that endpoint from both the left and right sides.

3. Can a piece-wise function have an infinite number of pieces?

Yes, a piece-wise function can have an infinite number of pieces. This is often the case when the function has a different equation for each individual point on the number line.

4. What is the purpose of showing a piece-wise function is continuous?

Showing that a piece-wise function is continuous is important because it ensures that the function is well-defined and does not have any gaps or jumps in its graph. This is necessary for many applications in mathematics and science, where continuity is a fundamental concept.

5. Are there any specific techniques for showing piece-wise functions continuous?

Yes, there are specific techniques for showing piece-wise functions continuous. One common technique is using the epsilon-delta definition of continuity, where you must show that for any given epsilon value, there exists a corresponding delta value that satisfies the definition of continuity. Another technique is using the intermediate value theorem to show that the function takes on all values between two given points on the number line.

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