Question

Suppose \(f\) is integrable on \([a, b], \alpha=\inf _{a \leq x \leq b} f(x),\) and \(\beta=\sup _{a \leq x \leq b} f(x)\) Let \(g\) be continuous on \([\alpha, \beta]\). Show that the composition \(h=g \circ f\) is integrable on \([a, b]\).

Short answer

Question: Given that the function \(f:[a, b] \to [\alpha, \beta]\) is integrable and that \(g:[\alpha, \beta] \to\) is continuous, show that the composition function \(h: x \mapsto g(f(x))\) is integrable on \([a, b]\). Answer: By defining a new function \(g' = g \circ f\) and showing that it is continuous on \([a, b]\), we find the difference between the upper and lower sums of \(h\). Since the difference converges to zero as the limit approaches infinity, we can conclude that the composition function \(h = g \circ f\) is integrable on \([a, b]\).

Step-by-step solution

In order to make the relationship between the functions \(f\) and \(h\) more apparent, let us define \(g' = g \circ f\). This means \(g'(x) = g(f(x))\) for all \(x \in [a, b]\). Notice that this new function \(g'\) is equivalent to the function \(h\) from our original problem. #Step 2: Show that \(g'(x)\) is continuous on \([a, b]\)#

We know that \(g\) is continuous on \([\alpha, \beta]\). Since \(\alpha \le f(x) \le \beta\) for all \(x \in [a, b]\), it follows that \(g'(x) = g(f(x))\) is also continuous on \([a, b]\). This is because the composition of continuous functions is also continuous. #Step 3: Calculate the difference between the upper and lower sums of \(h\)#

For the function \(h = g' = g \circ f\), we need to find partitions \(P\) of \([a, b]\) such that the difference between its upper and lower Riemann sums, \(U(h, P)\) and \(L(h, P)\), is less than \(\varepsilon\), where \(\varepsilon > 0\). Recall that for functions \(f(x)\), the difference between its upper and lower sums is given by \(U(f, P) - L(f, P) = \sum_{i=1}^n M_i \Delta x_i - \sum_{i=1}^n m_i \Delta x_i\), where \(M_i\) and \(m_i\) are the supremum and infimum, respectively, of the function \(f(x)\) on the interval \([x_{i-1}, x_i]\), and \(\Delta x_i = x_i - x_{i-1}\). Now let's consider the upper and lower sums of \(h(x) = g'(x)\). We will have \(U(h, P) - L(h, P)\) = \(\sum_{i=1}^n G_i \Delta x_i - \sum_{i=1}^n g_i \Delta x_i\), where \(G_i\) and \(g_i\) are the supremum and infimum, respectively, of the function \(h(x)\) on the interval \([x_{i-1}, x_i]\). Using the fact that \(g'(x)\) is continuous on \([a, b]\), we can invoke the Extreme Value Theorem, which says that \(h(x)\) attains both its maximum and minimum within each partition. Hence, the upper and lower sums will have the same value in the limit, which means the function \(h(x)\) is integrable. #Step 4: Demonstrate that the composition function \(h=g\circ f\) is integrable on \([a, b]\)#

Combining the conclusions from the previous steps, we can now demonstrate that \(h=g \circ f\) is integrable on \([a, b]\). We have shown that \(h(x) = g'(x)\) is continuous on \([a, b]\) and the difference between the upper and lower sums is given by \(U(h, P) - L(h, P) = \sum_{i=1}^n G_i \Delta x_i - \sum_{i=1}^n g_i \Delta x_i\). To overturn the case when \(h\) attains its supremum and infimum, we can find a partition of \([a, b]\) for which \(U(h, P) - L(h, P) < \varepsilon\), where \(\varepsilon > 0\). Since this property holds for all \(\varepsilon > 0\), we can conclude that the composition function \(h = g \circ f\) is integrable on \([a, b]\). This completes the proof.

Key concepts

Riemann Integral

The Riemann integral is a fundamental concept in calculus used to calculate the area under a curve over an interval. It is defined by taking the limit of sums of areas of rectangles (Riemann sums), as the width of the rectangles becomes infinitesimally small. To be more specific, for a function to be Riemann integrable on an interval \[a, b\], given any positive number \(\epsilon\), there must exist a partition \(P\) of \[a, b\] such that the difference between the upper sum \(U(f, P)\) and the lower sum \(L(f, P)\) is less than \(\epsilon\). This criterion ensures the function's behavior is sufficiently tame over the interval; variances in its values are not excessively large. A Riemann integrable function is guaranteed to have an integral that provides the 'total accumulated value' across the interval, which can represent area, probability, or other physical and mathematical concepts.

Continuous Function

Continuous functions are the bedrock of calculus. These functions have no sudden jumps or breaks and can be drawn without lifting a pencil off the paper. Formally, a function \(f\) is continuous at a point \(x_0\) if for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that whenever \(x\) is within \(\delta\) of \(x_0\), \(f(x)\) is within \(\epsilon\) of \(f(x_0)\). In simpler terms, as \(x\) gets arbitrarily close to \(x_0\), \(f(x)\) gets arbitrarily close to \(f(x_0)\), ensuring no abrupt changes occur. Continuity over an interval is a crucial property for integration; since integrals measure accumulation, the function being integrated must not 'jump around' to have a meaningful integral.

Composition of Functions

The composition of functions is like a mathematical relay race, where the output of one function becomes the input of another. Given two functions \(f\) and \(g\), the composition \(g \circ f\) is defined by \(g(f(x))\) for all \(x\) in the domain of \(f\) for which \(f(x)\) is in the domain of \(g\). If \(f\) maps every \(x\) to a value in its range, and \(g\) is ready to take that value and produce its own output, the composition will work smoothly. The concept is crucial in our exercise because showing that a composition is integrable involves understanding how the continuity of \(g\) interacts with the values produced by \(f\).

Supremum and Infimum

Supremum and infimum are like the 'bookends' of a function's values on a given interval. The supremum (often called the 'least upper bound') is the smallest value that is greater than or equal to every value of the function on that interval. Conversely, the infimum (or 'greatest lower bound') is the greatest value that is less than or equal to every value of the function. In Riemann integration, they play a critical role; for each subinterval in a partition, the supremum and infimum dictate the heights of the rectangles used to approximate the area under the curve. Understanding these concepts is necessary to grasp why a function's integrability implies that for any \(\epsilon > 0\), we can find partition rectangles that collectively differ from the integral by less than \(\epsilon\).

Extreme Value Theorem

The Extreme Value Theorem is like a guarantee that a continuous function won't leave you hanging. It makes a bold claim: if a function is continuous on a closed interval, then it must reach both its highest and lowest values somewhere within that interval. This theorem is critical when dealing with Riemann integrals. When we talk about the upper and lower sums in Step 3 of the solution, the theorem assures us that the supremum (the maximum) and infimum (the minimum) are actual values of the function on every subinterval. With this guarantee, we can confidently say that a continuous function will be integrable because we can 'trap' its area with these maxima and minima, ensuring that the upper and lower sums get closer and closer as the partition gets finer.