This document discusses continuity and discontinuities in functions, defining continuous functions as smooth curves where the domain includes all real numbers. It presents tests for continuity, requiring a function to be defined at a point, have an existing function value, and approach the same y-value from both sides. Critical points and extrema are defined as maximums where a function changes from increasing to decreasing, and minimums where it changes from decreasing to increasing. Rational functions are described as the quotient of two polynomial functions, with limited domains and possible vertical asymptotes.

1. ContinuityPrecalculus

2. Continuous functionsGraphically, a smooth, solid curve or line.Domain is not limited. All Real numbers.

3. DiscontinuitiesInfiniteJumpPoint

4. Continuity TestThe function is defined at c. f(c) exists.

5. Continuity TestThe function is defined at c. f(c) exists.The function approaches the same y value on the left and rightsides of x = c.

6. Continuity TestThe function is defined at c. f(c) exists.The function approaches the same y value on the left and rightsides of x = c.The y value that the function approaches from each side is f(c).

7. Continuity on an intervalA function f(x) is continuous on an interval IFF it is continuous at each number x in the interval.

8. Critical Points and ExtremaCritical points are the points on a graph at which a line drawn tangent to the curve is horizontal or vertical.

9. Critical Points and ExtremaMaximum is where the function changes from increasing to decreasing.

10. Critical Points and ExtremaMaximum is where the function changes from increasing to decreasing.Minimum is where the function changes from decreasing to increasing.

11. Critical Points and ExtremaMaximum is where the function changes from increasing to decreasing.Minimum is where the function changes from decreasing to increasing.Point of inflectionis where the graph changes its curvature.

12. Rational functionsThe quotient of two polynomial functions

13. Rational FunctionsLimited domains

14. Rational FunctionsVertical asymptotes or holesLimited domains