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Calculus Notes for 1.4 ***Memorize***Definition of Continuity at a Point:A function f is continuous at c if the following 3 conditions are met. 1. f(c) is defined. 2. lim f ( x ) exists 3. lim f ( x ) = f(c) x c x c Continuity on an open interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. If a function f is defined on an interval I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. A discontinuity is said to be removable if f can be made continuous by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is said to be nonremovable. Theorem 1.10 The Existence of a Limit: Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim f ( x) L and lim f ( x) L x c x c A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and lim f ( x) f (a) and lim f ( x) f (b) x b x a Definition of Continuity on a Closed Interval A function is continuous on a closed interval [a,b] if it is continuous on the open interval (a,b) and lim+ 𝑓(𝑥) = 𝑓(𝑎) 𝑥→𝑎 and lim 𝑓(𝑥) = 𝑓(𝑏) 𝑥→𝑏 − Intermediate Value Theorem: If f is continuous on the closed interval [a,b], f(a) ≠ f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k. Properties of Continuity: If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c: 1. bf 2. f±g 3. fg 4. f/g Example 1: Discuss the continuity of each function: this means to 1) tell any x-values where the function is discontinuous, 2) tell if each discontinuity is removable or non-removable and 3)tell which reason from the definition of continuous is not met a) 𝑓(𝑥) = b) 𝑓(𝑥) = 1 𝑥 𝑥 2 −1 𝑥−1 c) 𝑓(𝑥) = { 𝑥 + 1 𝑓𝑜𝑟 𝑥 ≤ 0 𝑥 2 + 1 𝑓𝑜𝑟 𝑥 > 0 d) y = sin(x) Example 2: 𝑓(𝑥) = √4 − 𝑥 2 . Find lim 𝑓(𝑥) 𝑥→−2+ The Greatest Integer function: ⟦𝑥⟧ = greatest integer n such that n≤x Example 3: f(x) = ⟦𝑥⟧ lim 𝑓(𝑥) 𝑥→0+ lim 𝑓(𝑥) 𝑥→0−