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:
A function is continuous over an interval if, for every point within that interval, the function is defined, the limit of the function as it approaches that point exists, and the limit equals the function's value at that point. In other words, there are no breaks, jumps, or holes in the graph of the function over that interval.
To prove that g(x) is continuous over the interval (-∞,-2), you need to:1. Ensure that g(x) is defined for all x in (-∞,-2).2. Show that the limit of g(x) as x approaches any point c in (-∞,-2) exists.3. Verify that this limit equals g(c) for all c in (-∞,-2).
Common methods to prove the continuity of a function include:1. Using the definition of continuity (limit definition).2. Applying known theorems, such as the Intermediate Value Theorem or the fact that polynomials and rational functions (where the denominator is non-zero) are continuous.3. Checking the continuity of component functions if g(x) is composed of simpler functions.
Potential challenges include:1. Identifying points where g(x) might not be defined or might have discontinuities.2. Calculating limits, especially if g(x) involves complex expressions.3. Ensuring that all points in the interval (-∞,-2) are accounted for, particularly if the function has different forms over different subintervals.
No, a function cannot be continuous over an interval if it has a discontinuity at any point within that interval. Continuity requires that the function be continuous at every point in the interval. If there is even one point where the function is not continuous, then the function is not continuous over that interval.