Math_137_Winter_2010_Week_4_Notes

Math_137_Winter_2010_Week_4_Notes - Math 137 Week 4 Notes...

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Math 137 Week 4 Notes – Dr. Paula Smith I. Continuity A. Definitions: Suppose f is de ned on [a, b]. 1. f is continuous at x = c (a, b) if lim x c f(x) = f(c). a) f(c) must exist b) lim x c f(x) must exist c) they must equal each other 2. f is continuous from the right at x = a if lim x a+ f(x) = f(a). 3. f is continuous from the left at x = b if lim x b f(x) = f(b). 4. f is continuous on (a, b) if f is continuous at every point c (a, b). 5. f is continuous on [a, b] if f is continuous at every point c (a, b) and also at continuous from the right at a and continuous from the left at b. B. Continuity Laws. Suppose f and g are continuous on interval I. 1. Sum Law: f ± g is continuous on I. 2. Product Law: f · g is continuous on I. 3. Quotient Law: f is continuous on the subset of I where g 0. 4. Composition Law: If f is continuous on Ran g I, then f(g(x)) is continuous on the subset of I where the function is de ned. C. Remarks. 1. “Basic” functions such as polynomials, |x|, a x , log a x, x p , trig functions, arcsin x, arctan x are continuous at every point in their domains. This is not necessarily the same as continuous on the real-number line. 2. A function has a “removable” discontinuity at c if lim x c f(x) exists. A function has a “jump” discontinuity at c if lim x c f(x) does not exist. Geometrically, f is continuous at x = c if the graph y = f(x) has no break at x = c. 3. If f is continuous at x = c, then we can interchange the order of f and lim symbol: lim f(x)= f(lim x) = f(c) [“Substitution Law”] II. Intermediate Value Theorem (IVT) A. IVT . Suppose f is continuous on [a, b] and f(a)
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Math_137_Winter_2010_Week_4_Notes - Math 137 Week 4 Notes...

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