Continuous

Definition

A function is continuous if there are no breaks, holes, or jumps in its graph. This means you can draw the graph without lifting your pencil.

5 Must Know Facts For Your Next Test

1. Polynomial functions are always continuous everywhere on their domain.
2. A function is continuous at a point $x = c$ if $\lim_{{x \to c}} f(x) = f(c)$.
3. Continuity implies that small changes in $x$ result in small changes in $f(x)$.
4. If a polynomial has real coefficients, it will be continuous over all real numbers.
5. Rational functions may have discontinuities at points where the denominator equals zero.

Review Questions

• What condition must a function meet to be considered continuous at a point?
• Are polynomial functions always continuous? Explain why or why not.
• Why might a rational function not be continuous at certain points?

Related terms

Discontinuity: A point where a function is not continuous. This can occur due to holes, jumps, or vertical asymptotes.

Limit: The value that a function approaches as the input approaches some value. Limits are essential for defining continuity.

Asymptote: $y$-values that a graph approaches but never touches or crosses. Vertical asymptotes often indicate discontinuities in rational functions.