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
- A polynomial function is always continuous.
- Continuity at a point means the function's value at that point equals the limit of the function as it approaches that point.
- If a function is not continuous at a point but has a limit that exists, it may have a removable discontinuity.
- Rational functions can be discontinuous where their denominators are zero.
- 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?
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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.
