NOTES 1.5 Continuity on Intervals &amp; IVT.pdf - Notes 1.5 Continuity Continued IVT §1.5—Continuity Continued IVT A continuous function A

# NOTES 1.5 Continuity on Intervals & IVT.pdf - Notes 1.5...

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Notes 1.5: Continuity Continued & IVT Page 1 of 8 §1.5—Continuity Continued & IVT A continuous function A non-continuous function To review the 3-step definition of continuity at a point, A function ( ) f x is continuous at a point x c = if ( ( ( lim lim x c x c f x f c f x - + = = Example 1: Determine the values at which the function ( 29 2 1 , 2 3 2 5, 2 2 , 2 4 8 , 4 x x x x f x x x x x < - + - < = < is continuous, then find ( lim x f x →-∞ and ( lim x f x →∞ .
Notes 1.5: Continuity Continued & IVT Page 2 of 8 Continuity on an interval A function ( f x is continuous on an interval I , if and only if the function ( f x is continuous at every point in the interval I . More specifically, Continuity on an open interval A function ( f x is continuous on an open interval ( , a b if and only if the function ( f x is continuous at every point in the interval ( , a b . Additionally, Continuity on a closed interval A function ( ) f x is continuous on a closed interval [ ] , a b if it is continuous on the open interval ( , a b and if the endpoints exhibit the following: (i) lim ( ) ( ) x a f x f a + = and (ii) lim ( ) ( ) x b f x f b - = Example 2: The graph of ( f x is given below. Determine the largest intervals 0 x 3 for which the function f x ( is continuous.
Notes 1.5: Continuity Continued & IVT Page 3 of 8 Example 3: The graph of a familiar function ( f x is given below. Answer the following questions based on the graph of ( f x .

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