2.4 Continuity and its Consequences and 2.8 IVT

Tues Sept 15

Do Now

Find the errors in the following and explain why it’s wrong:

HW Review: p.80 #5 13 19 27 29 33 35

• 5) 1/2

• 13) -2/5

• 19) 1/5

• 27) 3

• 29) 1/16

• 33) let f(x) = 1/x and g(x) = -1/x

• 35) proof

Continuity - What does it mean?

A function is said to be continuous on an interval if its graph on that interval can be drawn without interruption, or without lifting your pencil.

• Holes and asymptotes are examples of discontinuous functions

Definition of continuous

• A function f is continuous at x = a when

• 1) f(a) is defined

• 2) exists

• 3)

• Otherwise, f is said to be discontinuous at x = a

One-Sided Continuity

• A function f(x) is called:– Left-continuous at x = c if

– Right-continuous at x = c if

What kind of functions are continuous?

• Polynomials

• Radical Functions on their domains

• Sin x and cos x

• Exponential functions

• Logarithmic functions on their domains

• Rational functions on their domains

Piecewise Functions

• These kind of functions are the big AP type of problems

More Continuous Functions

• Thm- Suppose that f and g are continuous at x = c. Then:– 1) kf(x) for any constant k– 2) is continuous at x = c– 3) is continuous at x = c– 4) is continuous at x = c if and discontinuous if g(c) = 0

More Continuous Functions

• Thm- If f(x) is continuous on an interval I with range R and its inverse exists, then its inverse is continuous with domain R

Composite Functions

• If g(x) is continuous at x = c, and f(x) is continuous at x = g(c), then f(g(x)) is also continuous at x = c

3 Types of Discontinuities

• Removable Discontinuity– Limit exists– F(x) is not equal to the limit– Can redefine function at discontinuity

• Jump Discontinuity– Left and right side limits do not agree– Cannot redefine

• Infinite Discontinuity– One or both of each sided limits is infinite

Intermediate Value Theorem

• Suppose f is continuous on the closed interval [a,b] and W is any number between f(a) and f(b). Then, there is a number c in [a,b] s.t.

• From every y on [f(a),f(b)], there must be a corresponding x value that takes it there.

Ex

• Prove that the equation sin x = 0.3 has at least one solution

Closure

• Journal Entry: What must be true for a function to be continuous? What is an example of a discontinuity? Which are removable or not?

• HW:• 2.4 #2, 3, 4, 19, 27, 35, 59, 63• 2.8 #1, 7, 13