In calculus, a discontinuity occurs when a function is not continuous at a particular point. There are primarily three types of discontinuities: removable, jump, and infinite. A removable discontinuity exists when the limit of the function exists at that point, but the function is either undefined or has a different value. A jump discontinuity occurs when the function approaches different values from the left and the right at that point. Finally, an infinite discontinuity (also known as a vertical asymptote) happens when the function approaches infinity (or negative infinity) as it approaches the point.
Understanding these distinctions is crucial for analyzing function behavior and applying calculus concepts effectively.
Match the following terms with their correct definitions:
| Term | Definition |
|---|---|
| 1. Removable Discontinuity | A. The function approaches different finite values from the left and right. |
| 2. Jump Discontinuity | B. The function approaches infinity or negative infinity. |
| 3. Infinite Discontinuity | C. The limit exists, but the function is undefined or has a different value. |
| 4. Continuous Function | D. A function that has no discontinuities. |
| 5. Point Discontinuity | E. Another name for removable discontinuity. |
Complete the following paragraph with the correct terms:
A _________ discontinuity can be "fixed" by redefining the function at a single point. A _________ discontinuity occurs when the left-hand limit and the right-hand limit exist but are not equal. An _________ discontinuity results in a vertical asymptote on the graph of the function.
Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. Does this function have a discontinuity? If so, what type of discontinuity is it, and how would you prove it mathematically?
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