Understanding Continuity and Limits in Calculus

1. September 7th, 2017 Continuity and One-Sided Limits (1.4)

2. I. Continuity at a Point on an Open Interval • Def: A function is continuous at a point c if • 1. f(c) is defined • 2. exists, and • 3. . A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on the entire real line is everywhere continuous.

3. If a function f is continuous on the open interval (a, b) except at point c, it is said to have a discontinuity at c. This discontinuity is removable if f can be made continuous by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is nonremovable.

4. f(c) is not defined Removable discontinuity at c.

5. Nonremovable discontinuity at c. (This is known as a jump discontinuity) Another type of nonremovable discontinuity is an asymptote. does not exist

6. Removable discontinuity at c.

7. Ex. 1: Find all values of x for which the function is discontinuous. Using the definition of continuity, justify why the function is not continuous at each given value of x. Also describe the discontinuity as a jump discontinuity, a hole, or an asymptote. • (a) • (b) • (c) • (d)

8. II. One-Sided Limits and Continuity on a Closed Interval One-Sided Limits are denoted by the following. means the limit as x approaches c from the right, and means the limit as x approaches c from the left.

9. Thm. 1.10: The Existence of a Limit:Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if and

10. Ex. 2: Find each limit (if it exists). If it does not exist, explain why.

11. Def: A function f is continuous on a closed interval [a, b] if it is continuous on the open interval (a, b) and and . We say that f is continuous from the right of a and continuous from the left of b.

12. Ex. 3: Give each of the intervals of x for which the following function is continuous.

13. III. Properties of Continuity • Thm. 1.11: Properties of Continuity: If b is a real number and f and g are continuous at x=c, then the following functions are also continuous at c. • 1. Scalar multiple: bf • 2. Sum and difference: • 3. Product: fg • 4. Quotient:

14. Functions that are Continuous at Every Point in their Domain: • 1. Polynomial functions • 2. Rational functions • 3. Radical functions • 4. Trigonometric functions

15. Thm. 1.12: Continuity of a Composite Function: If g is continuous at c and f is continuous at g(c), then the composite function given by is continuous at c.

16. IV. The Intermediate Value Theorem ***Thm. 1.13: The Intermediate Value Theorem: If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c)=k.

17. Ex. 4: Prove that the function has a value of x such that in the interval

18. Ex. 5: Water flows into a cylindrical tank continuously for 12 hours, as given by the table below. What is the least number of times within those 12 hours that the rate of the water flow is 10 gallons/minute? Justify your reasoning.