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. Continuity Precalculus

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

3. Discontinuities Infinite Jump Point

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

5. Continuity Test The 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 Test The 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 interval A function f(x) is continuous on an interval IFF it is continuous at each number x in the interval.

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

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

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

11. Critical Points and Extrema Maximum 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 functions The quotient of two polynomial functions

13. Rational Functions Limited domains

14. Rational Functions Vertical asymptotes or holes Limited domains