Proving g(x) is continuous over interval (-∞,-2)

In summary, when referring to the interval from negative infinity to -2, the notation -∞ < a < -2 or a ∈ (-∞, -2) is used and it conveys the same information. This notation typically includes a parenthesis to indicate an excluded endpoint and a bracket for an included endpoint. The use of "For" in a compound inequality is not necessary.
  • #1
1,169
132
Homework Statement
Please see below
Relevant Equations
Please see below
For number 18,
1676597607436.png

The solution is,
1676597636101.png

However, should they not write "For ## -∞ < a < -2##" since ##a ≠ -∞## (infinity is not a number)?

Many thanks!
 
Physics news on Phys.org
  • #2
##a## is a number, so can't be negative infinity.

In particular, ##a\in [-\infty,-2)## isn't really a meaningful thing outside of people abusing notation.
 
  • Like
Likes ChiralSuperfields
  • #3
Office_Shredder said:
##a## is a number, so can't be negative infinity.

In particular, ##a\in [-\infty,-2)## isn't really a meaningful thing outside of people abusing notation.
Thank you for your reply @Office_Shredder !

So a better notation than ## a < -2 ## is ## -∞ < a < -2##, correct?

Many thanks!
 
  • #4
Callumnc1 said:
Thank you for your reply @Office_Shredder !

So a better notation than ## a < -2 ## is ## -∞ < a < -2##, correct?

Many thanks!
No, the ##-\infty<a## notation conveys no additional information. ##a## and ##a>-\infty## are the same thing. You can include the negative infinity, but in no sense is it better here.
 
  • Like
Likes ChiralSuperfields
  • #5
Callumnc1 said:
However, should they not write "For −∞<a<−2"
##-\infty < a < -2## and ##a \in (-\infty, -2)## are two different notations that say exactly the same thing. In most of the books I've seen, interval notation, as in the 2nd example above, uses a parenthesis to indicate an endpoint that isn't included, and a bracket to indicate that an endpoint is included. I've seen other notations used, but these seem to be a lot rarer.

Also, I don't think any textbook would include "For" in a compound inequality. ##-\infty < a < -2## says everything that needs to be said.
 
  • Like
Likes ChiralSuperfields

1. How do you prove that g(x) is continuous over the interval (-∞,-2)?

To prove that g(x) is continuous over the interval (-∞,-2), we need to show that the limit of g(x) as x approaches any value within the interval (-∞,-2) is equal to the value of g(x) at that point. This can be done using the definition of continuity and the properties of limits.

2. What is the definition of continuity?

The definition of continuity states that a function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, there are no breaks or gaps in the graph of the function at that point.

3. Can you use the Intermediate Value Theorem to prove continuity?

Yes, the Intermediate Value Theorem can be used to prove continuity. This theorem states that if a function is continuous over a closed interval [a,b] and takes on values y1 and y2 at points a and b, then it must also take on every value between y1 and y2 at some point within the interval [a,b].

4. What are the properties of limits that can be used to prove continuity?

The properties of limits that can be used to prove continuity include the sum, difference, product, and quotient rules. These rules state that if the limits of two functions exist, then the limit of their sum, difference, product, or quotient also exists and is equal to the sum, difference, product, or quotient of their limits, respectively.

5. What is the importance of proving continuity over an interval?

Proving continuity over an interval is important because it ensures that the function is well-behaved and has no breaks or gaps within that interval. This allows us to make accurate predictions and calculations using the function. Additionally, continuity is a fundamental concept in calculus and is necessary for understanding more complex concepts such as derivatives and integrals.

Suggested for: Proving g(x) is continuous over interval (-∞,-2)

Back
Top