Continuity and One-Sided Limits

Continuity and One-Sided Limits Lesson 2.4

Don't let this Happen to you!

Intuitive Look at Continuity • A function withoutbreaks orjumps • The graph can bedrawn 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

if x ≠ 1 and F(x) = 4 if x = 1 if x ≠ 1 and g(x) = 6 if x = 1 if x ≠ 1 and h(x) = 4 if x = 1 Which of These is Dis/Continuous? • When x = 1 … why or not 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 = cThen … • is continuous (where s is a constant) • f(x) + g(x) is continuous • 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?

c 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) 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 • Page 99 • Exercises 51 – 75, 85 – 105 odd

Continuity and One-Sided Limits