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

1. A function f(x) is continuous on an interval [a, b] if it is continuous at every point within [a, b].
2. If a function is continuous on a closed interval [a, b], then it must also be continuous on the open interval (a, b).
3. The Intermediate Value Theorem can be applied to functions that are continuous over an interval.
4. Polynomials are examples of functions that are continuous over all real numbers.
5. 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.

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.