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Unformatted text preview: 2.6 Continuity Intermediate Value Theorem (IVT) for Continuous Func- tions THEOREM 12 - IVT for Continuous Functions A function y = f ( x ) that is continuous on a closed interval [ a, b ] takes on every value between f ( a ) and f ( b ). In other words, if y is any value between f ( a ) and f ( b ), then y = f ( c ) for some c in [ a, b ]. Geometrically, the Intermediate Value Theorem says that any horizontal line y = y crossing the y-axis between the numbers f ( a ) and f ( b ) will cross the curve y = f ( x ) at least once over the interval [ a, b ]. A Consequence for Graphing: Connectivity The graph of a function continuous on an interval cannot have any breaks over the interval. It is connected- a single, unbroken curve. No jumps! The graph of a continuous function f can be sketched over its domain in one continuous motion without lifting the pencil. 1 A Consequence for Root Finding A solution fo the equation f ( x ) = 0 is a root of the equation or zero of the function....
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