Continuous

Definition

A function is continuous if there are no breaks, holes, or jumps in its graph. In other words, you can draw the entire graph without lifting your pen.

5 Must Know Facts For Your Next Test

1. A polynomial function is always continuous.
2. Continuity at a point means the function's value at that point equals the limit of the function as it approaches that point.
3. If a function is not continuous at a point but has a limit that exists, it may have a removable discontinuity.
4. Rational functions can be discontinuous where their denominators are zero.
5. The Intermediate Value Theorem applies to continuous functions over an interval.

Review Questions

• What does it mean for a polynomial function to be continuous?
• How do you determine if a rational function is continuous at a specific point?
• What theorem guarantees that a continuous function will take on every value between two points?

Related terms

Intermediate Value Theorem:

If a function is continuous on [a, b], then it takes on every value between f(a) and f(b).

Removable Discontinuity: A point at which a graph is not connected but can be made continuous by redefining the function.

Limit:

The value that a function approaches as the input approaches some value.