1.4 Continuity and one-sided limits

1. 1.4 Continuity and one-sided limits "Mathematics -- the subtle fine art." -- Jamie Byrnie Shaw

2. Objective • To determine the continuity of functions • To find one-sided limits

3. Top 10 excuses for not doing your math homework • #10. Galileo didn't know calculus; what do I need it for? • #9. "A math addict stole my homework. • #8. I'm taking physics and the homework in there seemed to involve math, so I thought I could just do that instead. • #7. I have the proof, but there isn't room to write it in the margin. • #6. I have a solar powered calculator and it was cloudy.

4. Cont… • #5. I was watching the World Series and got tied up trying to prove that it converged. • #4. I could only get arbitrarily close to my textbook. I couldn't actually reach it. (I reached half way, and then half of that, and then ...) • #3. I couldn't figure out whether i am the square root of negative one or i is the square root of negative one. • #2. It was Einstein's birthday and pi day and we had this big celebration! (This only works for March 14) • #1. I accidentally divided by zero and my paper burst into flames.

5. Intuitive approach

6. Formal definition • A function is continuous at c if: • 1. f(c) is defined ( f(c) exists ) • 2. • 3.

7. Simply put.. • A function is continuous if you can draw it without picking up your pencil • Can be continuous over open or closed intervals, or the entire function

8. Examples • When are the following functions continuous?

9. Discontinuities • If a function is discontinuous at a point, the discontinuity may be removable or non-removable depending upon whether the limit of the function exists at the point of discontinuity • In other words: • Removable – holes • Non-removable- breaks or asymptotes

10. Removable discontinuities • Denominators of fractions that factor with the numerators • Ex • Holes occur where you can take the limit but the actual value does not exist

11. Non-removable discontinuities • Asymptotes- when the denominator is zero • Ex: • This is when the actual value and the expected value or limit do not exist

12. Properties of continuity • If b is a real number, and f and g are continuous at x = c, then… • 1. Scalar multiple: bf(x) is continuous at c • 2. Sum and difference: f + g is continuous at c • 3. Product: fg is continuous at c • 4. Quotient: f/g is continuous at c, as long as g(c) does not equal 0

13. Left and right limits • We can look at limits from just the left or just the right • Right: Left:

14. Greatest integer function • Graph • Find:

15. Remember… • Three things must happen for a limit to exist • The limit from the right exists • The limit from the left exists • The limit from the right equals the limit from the left

16. Intermediate Value Theorem (IVT) • If f is continuous on [a,b] and k is a number between f(a) and f(b) then there is a c such that if a < c< b then f(c)= k