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 # ) 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, #1, 7, 13