Piecewise Continuity Problem: Figure 9-10n shows the graph of Figure 9-10n Suppose that

Chapter 9, Problem 27

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Piecewise Continuity Problem: Figure 9-10n shows the graph of Figure 9-10n Suppose that you are to evaluate Although the integrand is discontinuous on the closed interval [1, 3], there is only a step discontinuity at x = 2. The integrand is continuous everywhere else in [1, 3]. Such a function is said to be piecewise-continuous on the given interval. In this problem you will show that a piecewise-continuous function is integrable on the given interval Definition: Piecewise ContinuityFunction f is piecewise-continuous on theinterval [a, b] if and only if there is a finitenumber of values of x in [a, b] at which f(x)is discontinuous, the discontinuities areeither removable or step discontinuities, and f is continuous elsewhere on [a, b]. a. Write the integral as the sum of twointegrals, one from x = 1 to x = 2 and theother from x = 2 to x = 3.b. Both integrals in part a are improper. Writeeach one using the correct limit terminology.c. Show that both integrals in part b converge.Observe that the expression |x 2|/(x 2)equals one constant to the left of x = 2 and adifferent constant to the right. Find the valueto which the original integral converges.d. Explain why this property is true Property: Integrability ofPiecewise-Continuous FunctionsIf function f is piecewise-continuous onthe interval [a, b], then f is integrable on[a, b]. e. True or false: A function is integrable onthe interval [a, b] if and only if it iscontinuous on [a, b]. Justify your answer.

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