Notes 2.3 – Continuity

I. Intro: A.) A continuous function has no breaks in its domain.

B.) Def- A function f is continuous at a point x = c 1.) 2.) 3.)

II. One-sided Continuity A.) A function f can be continuous at a left(a) or a right(b) endpoint of its domain if

B.) Visual Representations: 1.) Fig Continuous on [0, 4] except at x = 1 and 2

Cont. from right at x = a acb Cont. from left at x = b 2-sided Cont. at x = c

III. Types of Discontinuity A.) Def- Any point c where f is not continuous, f is called discontinuous. B.) Removable: Hole

C.) Jump D.) Infinite One-sided limits exists  diff. values x  c f  ±∞

E.) Oscillating Only B.) can be repaired!!! Oscillates too much to have a limit as x  c.

IV. Examples Determine where each of the following are discontinuous and what type of discontinuity is displayed. A.) B.) C.) None Infinite at x = 2 Jump at x = 1

IV. Examples D.) E.) Disc. At x = 1, removable Disc. At x = 3, removable What value would you give f(x) at x = 3 to make it continuous?

V. Removing a Discontinuity To remove a discontinuity, we will make an extended function. We will determine the value that will make the function continuous at the point of discontinuity by taking the limit of the function. Finally, we redefine the function, assigning the limit value to the function at the point of discontinuity.

IV. Theorems on Continuity A.) Polynomial functions are continuous everywhere. B.) If f(x) and g(x) are cont. at x = c then 1.) f(x) + g(x) is cont. at x = c. 2.) f(x) (g(x)) is cont. at x = c. 3.) f(x) /g(x) is cont. at x = c, if g(c) ≠ 0.

C.) Rational functions are cont. everywhere in their domain. D.) If f(x) and g(x) are cont. at x = c then 1.) f(g(x)) is cont. at x = c. 2.) g(f(x)) is cont. at x = c. ex.)

E.) Extreme Value Thm. – Every cont. fn. On [a, b] has a maximum and a minimum on [a, b]. F.) IntermediateValue Thm. – If f is a cont. fn. on [a, b] and c is any value between f(a) and f(b) inclusive, then there exists at least one value of x in [a, b] such that f(x) = c. What it really means- Continuous functions on [a, b] take on EVERY value between f(a) and f(b).

G.) Corollary to IntermediateValue Thm. – If f is a cont. fn. on [a, b] and f(a) and f(b) have opposite signs, then there exists at least one solution to f(x) = 0 in (a, b). H.) Application of F and G : Does have a solution on [1, 2]?