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

1. 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) $\lim_{{x \to c}} f(x) = f(c)$.
2. If any of the three conditions for continuity at a point are not met, the function is not continuous at that point.
3. Polynomials are continuous at every point in their domain.
4. Rational functions are continuous at every point where the denominator is not zero.
5. 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 $\lim_{{x \to c}} f(x) \neq f(c)$.

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 $| f(x)-L | < \epsilon$.