lecture8_1 - 2.6 Continuity Intermediate Value Theorem(IVT for Continuous Functions THEOREM 12 IVT for Continuous Functions A function y = f(x that is

# lecture8_1 - 2.6 Continuity Intermediate Value Theorem(IVT...

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2.6 ContinuityIntermediate Value Theorem (IVT) for Continuous Func-tionsTHEOREM 12 - IVT for Continuous FunctionsA functiony=f(x) that is continuous on a closed interval [a, b] takeson every value betweenf(a) andf(b).In other words, ify0is any valuebetweenf(a) andf(b), theny0=f(c) for somecin [a, b].Geometrically, the Intermediate Value Theorem says that any horizontal liney=y0crossing they-axis between the numbersf(a) andf(b) will cross thecurvey=f(x) at least once over the interval [a, b].A Consequence for Graphing: ConnectivityThe graph of a function continuous on an interval cannot have any breaksover the interval. It isconnected- a single, unbroken curve. No jumps!”The graph of a continuous functionfcan be sketched over its domain inone continuous motion without lifting the pencil.”1
A Consequence for Root FindingA solution fo the equationf(x) = 0 is arootof the equation orzeroof thefunction.