Question

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

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

Step-by-step solution

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.

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.

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.