Continuity over an interval
Definition
Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
5 Must Know Facts For Your Next Test
- A function f(x) is continuous on an interval [a, b] if it is continuous at every point within [a, b].
- If a function is continuous on a closed interval [a, b], then it must also be continuous on the open interval (a, b).
- The Intermediate Value Theorem can be applied to functions that are continuous over an interval.
- Polynomials are examples of functions that are continuous over all real numbers.
- To prove continuity over an interval, show that the limit as x approaches any point c within the interval equals the function value at c.
Review Questions
- What does it mean for a function to be continuous on an interval?
- How can you verify if a given function is continuous on the interval [a, b]?
- Give an example of a type of function that is always continuous over any real number.
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Related terms
Limit: The value that a function approaches as the input approaches some value.
Intermediate Value Theorem: If f(x) is continuous on [a, b] and k is between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = k.
Discontinuity: A point at which a mathematical function is not continuous.
