1. 2.4 Continuity and its Consequences Thurs Sept 17 Do Now Find the errors in the following and explain why it’s wrong:

2. 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

3. 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

4. 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

5. One-Sided Continuity A function f(x) is called: –Left-continuous at x = c if –Right-continuous at x = c if

6. 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

7. Piecewise Functions These kind of functions are the big AP type of problems

8. 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

9. 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

10. 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

11. 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

12. 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: p.88-89 #1, 3-5, 17-33 odds, 55 57 59 63 65

13. Continuity Cont’d Fri Sept 18 Do Now Is the function continuous at the following points? 1)X = 3 2)X = 4

14. HW Review: p.88-89 #1, 3-5, 17-33 odds, 55 57 59 63 65 1) RC@1 nether@3 LC@5 27) x = 2, jump, LC 3) X = 3 redefine g(3) = 429) t = (2n+1)pi/4, n = int 4) C = 1, redefine g(1) = 331) continuous for all 5) Omgicantfitthishere33) x = 0, inf, neither 17) X = 0, inf, neither55) show right lim = left 19) X = 1, inf, neither57) c = 5/3 21) Even ints, jump, RC 59) a = 2, b = 1 23) X = 1/2, inf, neither63) graph 25) Continuous for all x65) graph

15. Classwork Side 1(p.53) #3, 4 Side 2(p.153) #21 22 23 24 25

16. Closure Exit pass: Find all discontinuities of For each discontinuity, state the type, whether it is left/right continuous, and if removable, redefine it so it is continuous HW: none or finish worksheet 2.3-2.5 quiz soon