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 $\lim_{{x \to c}} f(x) = f(c)$.
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A polynomial function is always continuous for all real numbers.
For a function to be continuous at a point, it must be defined at that point.
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 $f(c) = 0$.
Rational functions can be discontinuous where their denominators are zero.
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.