1. 1.4 Continuity and One-Sided Limits Main Ideas Determine continuity at a point and continuity on an open interval. Determine one-sided limits and continuity on a closed interval. Use properties of continuity. Understand and use the Intermediate Value Theorem.

2. Informal Definition of continuity Graph with only one pen stroke. Predictable http://www.calculus-help.com/funstuff/phobe.html Continuity

3. Definition of Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1.f(c) is defined. (a point exists) 2.Lim f(x) exists. ( no gap or jump in the graph) 3.Lim f(x) = f(c). (no hole in the graph) Continuity on an Open Interval: 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 an the entire real line (- ∞, ∞ ) is everywhere continuous.

4. Discontinuous If the function fails one of the conditions for continuity then it is discontinuous at c and any open interval that contains c. There are two types of discontinuities; Removable (Hole) If f can be made continuous by appropriately redefining f(c). Nonremovable (Asymptote or jump) When f(c) can not be defined.

5. Identify if the function continuous? If not label the discontinuity.

6. One-sided limits Limit from the right of c. lim f(x) = L – As you approach c from values that are larger than c. Limit from the left of c. lim f(x) = L – As you approach c from values that are smaller than c. Remember lim f(x) = L exists only if the one sided limits are equal to each other!

8. One-sided limits also let us expand upon the definition of continuity to include a closed interval. Continuity on a closed interval A function is continuous on a closed interval [a,b] if it is continuous on the open interval (a,b) and the function is continuous from the right at the left-hand endpoint and continuous from the left at the right-hand endpoint.

9. What kind of function(s) are continuous over an open interval? Polynomial What kind of function(s) are continuous over an open interval of their domain? Rational Radical Trigonometric What kind of function(s) do you need to use the definition of continuity to determine if they are continuous? Piece-wise Absolute value Composite

11. 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. How is the theorem used? 1.It is an existence theorem so it is used in proofs of many other mathematical concepts. 2.To locate zero’s of a function that are continuous.