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Chapter 4: Problem 92

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([a, b]\), then \(f\) is integrable on \([a, b]\).

Short Answer

Expert verified
The statement is true. All continuous functions on a closed interval are integrable on that interval.

Step by step solution

01

Understand the definitions

A function \(f\) is said to be continuous on a given interval if for every point in the interval, the limit of \(f\) as \(x\) approaches that point exists and is equal to the value of \(f\) at that point. A function is said to be integrable over an interval when the integral of the function over that interval exists.
02

Analyze the Statement

It's essential to analyze if the continuity of a function over a given interval necessarily implies that the function is also integrable on that interval. It is a well-known mathematical fact that all continuous functions are integrable. This fact is based on different properties of continuous functions, including that they are bounded and that they have a finite number of discontinuities on a closed interval, both of which are conditions for integrability.
03

Conclude the Statement

So, analyzing the definitions and the given statement, it can be concluded that the statement 'If a function \(f\) is continuous on \([a, b]\), then \(f\) is integrable on \([a, b]\)' is indeed true.

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Most popular questions from this chapter

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\tanh x, \quad a=0\)

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