Continuous at a point
Definition
A function $f(x)$ is continuous at a point $x = c$ if the limit of $f(x)$ as $x$ approaches $c$ exists, equals $f(c)$, and $f(c)$ is defined.
5 Must Know Facts For Your Next Test
- For a function to be continuous at a point $x = c$, it must satisfy three conditions: 1) $f(c)$ is defined, 2) $\lim_{{x \to c}} f(x)$ exists, and 3) $\lim_{{x \to c}} f(x) = f(c)$.
- If any of the three conditions for continuity at a point are not met, the function is not continuous at that point.
- Polynomials are continuous at every point in their domain.
- Rational functions are continuous at every point where the denominator is not zero.
- Removable discontinuities occur when $\lim_{{x \to c}} f(x)$ exists but does not equal $f(c)$ due to $f(c)$ being undefined or different.
Review Questions
- What are the three conditions required for a function to be continuous at a point?
- Explain why a rational function might fail to be continuous at certain points.
- Describe what happens to the continuity of a function if $\lim_{{x \to c}} f(x) \neq f(c)$.
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Related terms
Limit: The value that a function approaches as the input approaches some value.
Discontinuity: A point where a function is not continuous. Types include removable, jump, and infinite discontinuities.
$\epsilon - \delta$ Definition of Limit: $\lim_{{x \to c}} f(x) = L$ means for every $\epsilon >0$, there exists $\delta >0$ such that if $0 < | x-c | < \delta$, then $| f(x)-L | < \epsilon$.
