Continuity and One-Sided Limits Lesson 2.4

Don't let this Happen to you!

Intuitive Look at Continuity A function without breaks or jumps The graph can be drawn without lifting the pencil 

Continuity at a Point A function can be discontinuous at a point A hole in the function and the function not defined at that point A hole in the function, but the function is defined at that point

Continuity at a Point A function can be discontinuous at a point The function jumps to a different value at a point The function goes to infinity at one or both sides of the point, known as a pole

Definition of Continuity at a Point A function is continuous at a point x = c if the following three conditions are met f(c) is defined For this link, determine which of the conditions is violated in the examples of discontinuity x = c

"Removing" the Discontinuity A discontinuity at c is called removable if … If the function can be made continuous by defining the function at x = c or … redefining the function at x = c Go back to this link, determine which (if any) of the discontinuities can be removed

Which of These is Dis/Continuous? When x = 1 … why or not if x ≠ 1 and h(x) = 4 if x = 1 if x ≠ 1 and g(x) = 6 if x = 1 if x ≠ 1 and F(x) = 4 if x = 1 Are any removable?

Continuity Theorem A function will be continuous at any number x = c for which f(c) is defined, when … f(x) is a polynomial f(x) is a power function f(x) is a rational function f(x) is a trigonometric function f(x) is an inverse trigonometric function

Properties of Continuous Functions If f and g are functions, continuous at x = c Then … is continuous (where s is a constant) f(x) + g(x) is continuous is continuous f(g(x)) is continuous

One Sided Continuity A function is continuous from the right at a point x = a if and only if A function is continuous from the left at a point x = b if and only if a b

Continuity on an Interval The function f is said to be continuous on an open interval (a, b) if It is continuous at each number/point of the interval It is said to be continuous on a closed interval [a, b] if It is continuous at each number/point of the interval and It is continuous from the right at a and continuous from the left at b

Continuity on an Interval On what intervals are the following functions continuous?

Intermediate Value Theorem Given function f(x) Continuous on closed interval [a, b] And L is a number strictly between f(a) and f(b) Then … there exists at least one number c on the open interval (a, b) such that f(c) = L f(b) c L f(a) b a

Locating Roots with Intermediate Value Theorem Given f(a) and f(b) have opposite sign One negative, the other positive Then there must be a root between a and b a Try exercises 88, 90, and 92 : pg 100 b

Assignment Lesson 2.4A Page 98 Exercises 1 – 49 odd Lesson 2.4B