Continuity and One- Sided Limits (1.4) September 26th, 2012

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. is everywhere continuous.

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.

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

does not exist Nonremovable discontinuity at c.

Removable discontinuity at c.

Ex. 1: Discuss the continuity of each function. (a)(b)(c)(d)

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

Ex. 2: Find the limit (if it exists). If it does not exist, explain why. (a)(b)(c)

You try: (a)(b)

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

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.

Ex. 3: Discuss the continuity of the function on the closed interval [-1, 2]. on the closed interval [-1, 2].

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:

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

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.

Ex. 4: Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? (a)(b)

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.

Ex. 5: Explain why the function has a zero in the interval.