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1.4 Continuity and one-sided limits. "Mathematics -- the subtle fine art." -- Jamie Byrnie Shaw . Objective. To determine the continuity of functions To find one-sided limits. Top 10 excuses for not doing your math homework. #10. Galileo didn't know calculus; what do I need it for?

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## 1.4 Continuity and one-sided limits

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**1.4 Continuity and one-sided limits**"Mathematics -- the subtle fine art." -- Jamie Byrnie Shaw**Objective**• To determine the continuity of functions • To find one-sided limits**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.**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.**Formal definition**• A function is continuous at c if: • 1. f(c) is defined ( f(c) exists ) • 2. • 3.**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**Examples**• When are the following functions continuous?**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**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**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**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**Left and right limits**• We can look at limits from just the left or just the right • Right: Left:**Greatest integer function**• Graph • Find:**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**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

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