Thank you for your reply @Office_Shredder !Office_Shredder said:
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.Callumnc1 said:
##-\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.Callumnc1 said:
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.
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.
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].
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.
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.