1-4: Continuity and One-Sided Limits Objectives: Define and explore properties of continuity Discuss one-sided limits Introduce Intermediate Value Theorem ©2002 Roy L. Gover (roygover@att.net)

Definition f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

Examples Continuous Functions

Examples Discontinuous Functions Infinite discontinuity (non-removable) Jump Discontinuity (non-removable) Removable discontinuity

f(x) is continuous at x=c if and only if: Definition f(x) is continuous at x=c if and only if: 1. f (c) is defined …and 2. exists …and 3.

Examples Discontinuous at x=2 because f(2) is not defined x=2

Examples Discontinuous at x=2 because, although f(2) is defined, x=2

Definition f(x) is continuous on the open interval (a,b) if and only if f(x) is continuous at every point in the interval.

Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Continuous for all x

Try This Find the values of x (if any) where f is not continuous. Is the discontinuity removable? Discontinuous at x=o, not removable

Definition f(x) is continuous on the closed interval [a,b] iff it is continuous on (a,b) and continuous from the right at a and continuous from the left at b.

f(x) is continuous from the right at a a Example f(x) is continuous from the right at a f(x) is continuous on (a,b) a f(x) is continuous from the left at b f(x) b f(x) is continuous on [a,b]

Definition is a limit from the right which means x c from values greater than c

Definition is a limit from the left which means x c from values less than c

Example Find the limit of f(x) as x approaches 1 from the right:

Example Find the limit of f(x) as x approaches 1 from the left:

Example Find the limit of f(x) as x approaches 1:

Important Idea Theorem 1.10: exists iff

Try This Use the graph to determine the limit, the limit from the right & the limit from the left as x0.

Try This Use the graph to determine the limit, the limit from the right & the limit from the left as x1. x=1

Intermediate Value Theorem Theorem 1.13: If f is continuous on [a,b] and k is a number between f(a) & f(b), then there exists a number c between a & b such that f(c ) =k.

Intermediate Value Theorem f(a) k c f(b) b a

Intermediate Value Theorem an existence theorem; it guarantees a number exists but doesn’t give a method for finding the number. it says that a continuous function never takes on 2 values without taking on all the values between.

Example Ryan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.

Try This Use the Intermediate Value Theorem to show that has a zero in the interval [-1,1].

Solution therefore, by the Intermediate Value Theorem, there must be a f (c)=0 where

Lesson Close Tell me one thing you know about continuity and discontinuity.

Assignment 92/1-6 all,7-19 odd,27-49 odd,57,59,65, 81-84