Problem 32 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function and define \(g:[-b,-a] \rightarrow \mathbb{R}\) by \(g(t):=f(-t) .\) Show that \(L(g)=L(f)\) and \(U(g)=U(f) .\) Deduce that \(g\) is integrable on \([-b,-a]\) if and only if \(f\) is integrable on \([a, b]\) and in that case the Riemann integral of \(g\) is equal to the Riemann integral of \(f\). (Compare the proof of part (ii) of Proposition \(6.26 .\) )

Step-by-Step Solution

Answer

We have shown that for the bounded function \(f\) on \([a, b]\) and the function \(g(t) = f(-t)\) on \([-b, -a]\), their lower and upper sums are equal, that is \(L(g)=L(f)\) and \(U(g)=U(f)\). Consequently, we have deduced that \(g\) is Riemann integrable on \([-b, -a]\) if and only if \(f\) is integrable on \([a, b]\). In the case where both \(f\) and \(g\) are integrable, their Riemann integrals are equal, \(\int_{-b}^{-a} g(t) dt = \int_a^b f(x) dx\).

Step 1: Lower sum equality

We will analyze the equality of lower sums. Let \(P\) be a partition of \([a, b]\), and let \(P'\) be the partition of \([-b, -a]\) obtained by negating each element of \(P\). According to the definition of the lower sum, we have: \(L(f, P) = \sum_{i=1}^{n} \inf_{x \in [x_{i-1}, x_i]} f(x) \cdot (x_i - x_{i-1}) \) Similarly, for function \(g\) we have: \(L(g, P') = \sum_{i=1}^{n} \inf_{t \in [-x_i, -x_{i-1}]} g(t) \cdot (-x_{i-1} + x_i) \) By substituting \(g(t) = f(-t)\), we can write: \(L(g, P') = \sum_{i=1}^{n} \inf_{t \in [-x_i, -x_{i-1}]} f(-t) \cdot (-x_{i-1} + x_i) \) By changing the variable \(t = -x\), we have \(-t \in [x_{i-1}, x_i]\), and hence: \(L(g, P') = \sum_{i=1}^{n} \inf_{-t \in [x_{i-1}, x_i]} f(-t) \cdot (x_i - x_{i-1}) = L(f, P) \) This shows that the lower sums for \(f\) and \(g\) are equal.

Step 2: Upper sum equality

Similarly, we will analyze the equality of the upper sums. Using the same notation as in Step 1, we have: \(U(f, P) = \sum_{i=1}^{n} \sup_{x \in [x_{i-1}, x_i]} f(x) \cdot (x_i - x_{i-1}) \) \(U(g, P') = \sum_{i=1}^{n} \sup_{t \in [-x_i, -x_{i-1}]} g(t) \cdot (-x_{i-1} + x_i) \) By substituting \(g(t) = f(-t)\), we can write: \(U(g, P') = \sum_{i=1}^{n} \sup_{t \in [-x_i, -x_{i-1}]} f(-t) \cdot (-x_{i-1} + x_i) \) By changing the variable \(t = -x\), we have \(-t \in [x_{i-1}, x_i]\), and hence: \(U(g, P') = \sum_{i=1}^{n} \sup_{-t \in [x_{i-1}, x_i]} f(-t) \cdot (x_i - x_{i-1}) = U(f, P) \) This shows that the upper sums for \(f\) and \(g\) are equal.

Step 3: Integrability and integral equality

From Step 1 and Step 2, we have shown that \(L(g)=L(f)\) and \(U(g)=U(f)\) for any partitions \(P\) and \(P'\) of \([a, b]\) and \([-b, -a]\) respectively. Now, recall that a function is Riemann integrable if and only if the upper and lower integrals are equal, that is, if: \(\int_a^b f(x) dx = \lim_{||P|| \to 0} L(f, P) = \lim_{||P|| \to 0} U(f, P)\) Similarly, for function \(g\) we have: \(\int_{-b}^{-a} g(t) dt = \lim_{||P'|| \to 0} L(g, P') = \lim_{||P'|| \to 0} U(g, P')\) Since we have established the equality of lower and upper sums of \(f\) and \(g\) in Steps 1 and 2, we can deduce that: 1. If \(f\) is integrable on \([a, b]\), then \(g\) is integrable on \([-b, -a]\). 2. If \(g\) is integrable on \([-b, -a]\), then \(f\) is integrable on \([a, b]\). 3. If both \(f\) and \(g\) are integrable, then their Riemann integrals are equal, that is \(\int_{-b}^{-a} g(t) dt = \int_a^b f(x) dx\). This completes the proof.

Key Concepts

Integrable Functions

When we talk about integrable functions, we refer to functions for which it is possible to calculate the area under the curve over a certain interval. In the context of Riemann integration, a function is said to be integrable if the limit of the upper and lower sums of partitions of the interval converge to the same number as the partition gets infinitely fine.

This is akin to slicing the area under the curve into vertical strips, making the width of these strips narrower and narrower, and then approximating the area by summing up the areas of these strips. If the upper and lower estimates of the sum are equal in the limit (as the width of partitions approaches zero), the function's integral exists. In our exercise, we've analyzed such sums for function f and its mirror image g, concluding that if f is integrable over [a, b], so is g over [-b, -a].

Upper and Lower Sums

The concepts of upper and lower sums are critical to understanding Riemann integration. These sums are based on the idea of partitioning the interval and looking at rectangles that lie above and below the function's graph.

An upper sum is obtained by taking the supremum (least upper bound) of the function on each subinterval of a partition, and summing the products of these values and the subinterval widths. It's like building the tallest possible fences that still don't overshoot the curve. Conversely, a lower sum uses the infimum (greatest lower bound) values. It's like laying down the shortest fence sections beneath the curve. If both sums converge to the same value as the partitions get finer, the area under the curve—the integral—is precisely that common value. In our exercise, we've established the equality of both the upper and lower sums for function f and g. This equality is a foundational step to showing the Riemann integrability of g based on that of f, or vice versa.

Function Bounds

The exercise we tackled relies on understanding the bounds of a function. For a function to be Riemann integrable, the function must be bounded on the interval of integration. A function is bounded if there exists a real number that serves as a ceiling (upper bound) and a floor (lower bound) for its values.

For any partition of the interval, we look at these bounds within each subinterval to form our upper and lower sums. Finding the supremum and infimum values is central to determining these. In simple terms, no matter how much our function f wiggles within the interval [a, b], we can always say it does not exceed some specific heights or fall below certain levels. Our equations showed that the function g, created as a mirror image of f, honors these bounds in its own interval [-b, -a]. Therefore, boundedness is maintained from f to g, which is an essential condition for integrability.

Riemann Integrability Criterion

The Riemann integrability criterion is the final key topic we need to understand. It stipulates the conditions under which a function can be integrated using Riemann sums. To meet this criterion, the set of discontinuities of a function must be of measure zero, and the function should be bounded, as we previously discussed.

According to this criterion, if we can make the difference between the upper and lower sums arbitrarily small by refining the partition, the function is integrable. This agreed-upon limit is the value of the Riemann integral. In our exercise, we deduced that function g is integrable if and only if function f is, by showing that they share identical upper and lower sums for corresponding partitions. This mutuality of integrability based on the Riemann criterion underscores the concept that integrals fundamentally measure 'area', irrespective of the orientation of the function on the coordinate plane.