Question

Suppose \(f\) is a bounded real function on \([a, b]\), and \(f^{2} \in \mathscr{\text { on }}[a, b] .\) Does it follow that \(f \in \mathscr{R}\) ? Does the answer change if we assume that \(f^{3} \in \mathscr{R} ?\)

Short answer

Answer: No, knowing that \(f^2 \in \mathscr{R}\) or \(f^3 \in \mathscr{R}\) does not guarantee that \(f \in \mathscr{R}\). The information about the integrability of \(f^2\) or \(f^3\) is not sufficient to draw conclusions about the integrability of \(f\) itself.

Step-by-step solution

Understanding Riemann integrability

A function is considered Riemann integrable if its set of discontinuities has a measure of zero, i.e., it has "negligible" discontinuities. In simpler terms, if a function has too many "jumps," it is not Riemann integrable. The first part of the problem gives us information about \(f^2\), so let's examine this function.

Analyzing \(f^2 \in \mathscr{R}\)

Since we know that \(f^2 \in \mathscr{R}\), it means that the function \(f^2\) is Riemann integrable. However, this information alone does not tell us whether \(f\) itself is Riemann integrable. For example, consider the function f(x) = 1/x, which is not Riemann integrable on the interval \([-1, 1]\), but its square, f(x)^2 = 1/(x^2), is Riemann integrable on the same interval. Therefore, knowing that \(f^2 \in \mathscr{R}\) does not guarantee that \(f \in \mathscr{R}\).

Analyzing \(f^3 \in \mathscr{R}\)

Now, let's consider the case when \(f^3 \in \mathscr{R}\). Just like in the case of \(f^2\), knowing that \(f^3\) is Riemann integrable does not guarantee that \(f\) is Riemann integrable. To illustrate this, consider the function h(x) = x. The function is Riemann integrable, but its cube, h(x)^3 = x^3, is also Riemann integrable. However, the function g(x) = -1/x is not Riemann integrable on \([-1, 1]\), but its cube, g(x)^3 = -1/(x^3), is Riemann integrable on the same interval. Therefore, knowing that \(f^3 \in \mathscr{R}\) also does not guarantee that \(f \in \mathscr{R}\).

Conclusion

Based on our analysis, knowing that \(f^2 \in \mathscr{R}\) or \(f^3 \in \mathscr{R}\) does not guarantee that \(f \in \mathscr{R}\). The information about the integrability of \(f^2\) or \(f^3\) is not sufficient to draw conclusions about the integrability of \(f\) itself.

Key concepts

Bounded Real Functions

Bounded real functions are a core concept in mathematical analysis as they follow a fundamental property: no matter the input within their domain, the output of the function will never exceed a certain fixed value (the upper bound) or fall below another (the lower bound). For example, if we define a function f such that it takes any real number x and always produces a number less than or equal to 10 and greater than or equal to -10, then f is bounded on the real line.

When dealing with Riemann integration, which requires adding up infinitely many infinitesimal products to find the total area under a curve, it's important for the functions to be bounded. This is because unbounded functions can lead to integrals that diverge, being either infinitely large or not well-defined. For a bounded function like f^2 or f^3 mentioned in our exercise, the function values are contained within a specific range, hence such functions are good candidates for Riemann integrability, but that’s not the only condition as we’ll see in the next section about discontinuities.

Discontinuities of a Function

Discontinuities in a function are points at which the function is not 'smooth'. The behavior at these points can often be described as a 'jump', 'hole', or an 'infinite' discontinuity. When trying to understand Riemann integrability, these points play a crucial role.

For a function to be Riemann integrable, the set of all its discontinuity points must have a measure zero. In non-technical terms, the places where it ‘jumps’ should be so few and far between that, when you try to add all these up - the sum doesn't really count in the grand scale of the entire function.

Now, in the example of f^2 and f^3, even if these functions are bounded and integrable, they might possess discontinuities in such a way that the original function f doesn't meet the Riemann criterion. This could occur if squaring or cubing f removes or 'smooths out' its discontinuities. Thus, f might not be integrable even if f^2 or f^3 are.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These mathematical concepts are used to examine the properties and behaviours of real numbers, real-valued sequences and functions, and more.

In this particular exercise, Riemann integrability, a key concept in analysis, helps us understand which functions can be integrated, or visually speaking, which functions allow us to measure the area under their curve. While boundedness and limits are a part of the broader landscape of analysis, integrability adds another layer - not all functions that are bounded are integrable and not all functions that have limits at every point are necessarily continuous.

In our scenario, the exercise hints at something deeper than just ‘squaring’ or ‘cubing’ a function. It challenges one to truly understand how such operations affect a function’s integrability – a typical kind of problem that mathematical analysis is well-equipped to explore.