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2-3 continuity

2-3 continuity. Def: A function is continuous at: (1) an interior point if (2) a left endpoint if (3) a right endpoint if. a c b. How can a function be discontinuous?. Continuous Removable discontinuity. (limit exists). *to “remove” the discontinuity we

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2-3 continuity

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  1. 2-3 continuity

  2. Def: A function is continuous at: (1) an interior point if (2) a left endpoint if (3) a right endpoint if a c b How can a function be discontinuous? Continuous Removable discontinuity (limit exists) *to “remove” the discontinuity we can assign a value f (0) = limit

  3. JumpInfiniteOscillating DiscontinuityDiscontinuity Discontinuity limit DNE from the from the right left Strategies for removing a discontinuity: (1) Factor top & bottom (2) Something cancels (3) Use what’s left to assign value

  4. Def: A function is continuous on an intervaliff the function is continuous at every point on the interval Def: A function is a continuous functioniff the function is continuous at every point in its domain Note: This function has an infinite discontinuity at x = 0. However, we still consider the function to be a continuous function because x = 0 is not in its domain. Ex: Properties of Continuous Functions If f and g are continuous at x = c, then the following are also continuous at x = c: f + g (2) f – g (3) f g (4) k  f (5) f / g, g (c)  0 Also note that, composites of continuous functions are continuous

  5. Intermediate Value Theorem for Continuous Functions If a function is continuous on closed interval [a, b], then for some c in [a, b], y0 = f (c) where y0 is between f (a) & f (b). f (b) y0 f (a) a c b Ex 5) Is any real number exactly 1 less than it’s cube? Compute any such value accurate to three decimal places. x3 – 1 = x x3 – x – 1 = 0 Graph it  (1.325, 0)

  6. homework Pg. 76 #6, 18, 24, 30, 36, 42, 48, 54 Pg. 84 #2 – 44 (mult of 2+3n)

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