Continuous function

Definition

A continuous function is a function where small changes in the input result in small changes in the output. Mathematically, a function $f(x)$ is continuous at a point $x = c$ if $\lim_{{x \to c}} f(x) = f(c)$.

5 Must Know Facts For Your Next Test

1. A polynomial function is always continuous for all real numbers.
2. For a function to be continuous at a point, it must be defined at that point.
3. The Intermediate Value Theorem applies to continuous functions, stating that if $f(a)$ and $f(b)$ have opposite signs, then there exists at least one $c$ in $(a,b)$ such that $f(c) = 0$.
4. Rational functions can be discontinuous where their denominators are zero.
5. Continuity can be checked using limits: $\lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = f(c)$.

Review Questions

• What conditions must hold true for a function to be considered continuous at a specific point?
• How does the Intermediate Value Theorem relate to continuous functions?
• Explain why rational functions might not always be continuous.

Related terms

Polynomial Function:

A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Rational Function:

A ratio of two polynomial functions.

$\lim_{x \to c} f(x)$: The limit of $f(x)$ as $x$ approaches $c$, used to determine continuity.