Problem 22 - A Course in Calculus and Real Analysis

Question

2Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. Define \(G:[a, b] \rightarrow \mathbb{R}\) by $$ G(x):=\int_{x}^{b} f(t) d t . $$ Show that \(G\) is continuous on \([a, b] .\) Further, show that if \(f\) is continuous at \(c \in[a, b]\), then \(G\) is differentiable at \(c\) and \(G^{\prime}(c)=-f(c)\). (Hint: Propositions \(6.7,6.20\), and \(6.21\).)

Step-by-Step Solution

Answer

In summary, we have shown that the function $G(x):=\int_{x}^{b} f(t) d t$ is continuous on the interval $[a,b]$ by finding an upper and lower bound for the integral of the integrable function $f$. Furthermore, if $f$ is continuous at some point $c \in [a, b]$, we have demonstrated that $G$ is differentiable at $c$ and its derivative $G'(c)=-f(c)$, using the mean value theorem for integrals and the fundamental theorem of calculus.

Step 1: Show that \(G\) is continuous on \([a, b]\)

Let \(\epsilon > 0\). Since \(f\) is integrable on \([a, b]\), by definition there exists a partition \(P\) of \([a, b]\) such that $$ U(P,f)-L(P,f) < \epsilon, $$ where \(U(P,f)\) and \(L(P,f)\) are the upper and lower sums of \(f\) on the partition \(P\). Now, let \(x,y\in [a,b]\) with \(x

Step 2: Show that \(G\) is differentiable at \(c\) and \(G'(c)=-f(c)\) if \(f\) is continuous at \(c\in [a,b]\)

Since \(f\) is continuous at \(c\in [a,b]\), we have that for any \(\epsilon > 0\), there exists \(\delta > 0\) such that \(|f(t)-f(c)|<\epsilon\) for all \(t\) with \(|t-c|<\delta\). Now, let \(h>0\) be such that \(c+h\in [a,b]\) and \(h<\delta\). Then, by the mean value theorem for integrals (Proposition 6.20), there exists a point \(\xi \in(c,c+h)\) such that $$ \fbox{1}\hspace{2mm} \int_{c}^{c+h} f(t) dt = f(\xi) h. $$ Furthermore, by Proposition 6.21, we have $$ \fbox{2}\hspace{2mm} \lim_{h\to 0} \dfrac{f(\xi)h}{h} = f(c). $$ Taking the limit of equation \((1)\) as \(h\to 0\), we get $$ \lim_{h\to 0} G(c+h) - G(c) = -f(c)h. $$ Next, we divide both sides of the equation by \(h\) and take the limit as \(h\to 0\). /\. Since \( G\) is continuous, we have $$ \lim_{h\to 0} \dfrac{G(c+h) - G(c)}{h} = -f(c). $$ This implies that \(G'(c)=-f(c)\), meaning that \(G\) is differentiable at \(c\) and its derivative at \(c\) equals \(-f(c)\). This completes the proof.

Key Concepts

Continuity of Functions

Understanding the continuity of a function is crucial in calculus, as it forms the basis for more advanced concepts such as differentiation and integration. A function is said to be continuous on an interval if small changes in its input result in small changes in its output. Mathematically, a function \( G(x) \) is continuous on the interval \([a, b]\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x-y| < \delta\), it follows that \(|G(x)-G(y)| < \epsilon\).

This concept implies that the function has no abrupt changes or jumps within the interval. In the process of proving the continuity of \( G(x) = \int_{x}^{b} f(t) \, dt \), we showed that the difference \(|G(x) - G(y)|\) can be made arbitrarily small by choosing appropriate partitions, illustrating how integral properties lead to the conclusion that \(G(x)\) is indeed continuous.

Differentiation

Differentiation is the process of finding the derivative of a function, which indicates the rate at which the function's value changes with respect to its input. For the function \( G(x) = \int_{x}^{b} f(t) \, dt \), differentiation involves finding \( G'(x) \), the derivative of \( G \) at a specific point \( x \).

To determine differentiability, especially at a point \( c \) where \( f \) is continuous, we rely on the limit definition of the derivative: \( G'(c) = \lim_{h \to 0} \frac{G(c+h) - G(c)}{h} \). By using properties of \( f(t) \) when it is continuous at \( c \), and through meticulous limit calculations, it's observed that this process leads to \( G'(c) = -f(c) \). This is a fundamental result linking integration and differentiation, often highlighted by the Fundamental Theorem of Calculus.

Mean Value Theorem

The Mean Value Theorem (MVT) for integrals provides a powerful insight that bridges the functions' average value over an interval and its value at some point within the interval. When applied to functions like \( f(t) \), continuous on \([a, b]\), it states there exists a point \( \xi \in (a, b) \) where the integral of the function over \([a, b]\) equals the function's value at \( \xi \) multiplied by the interval's length.

In our exercise, this theorem helps in understanding why \( G \) is differentiable at a certain point. It offers a way to relate the change over an interval back to a single point, facilitating differentiation of \( G \). This relationship is particularly useful when proving \( G'(c) = -f(c) \), confirming the derivative's existence and its relation to function \( f \).

Riemann Integrals

Riemann integrals represent the classical method of defining an integral, focusing on the summation of areas under curves. This approach involves partitioning the domain into small intervals, calculating the sum of function values times small changes in intervals, and taking the limit of these sums as partitioning becomes finer.

For the function \( G(x) = \int_{x}^{b} f(t) \, dt \), verifying that \( f \) is Riemann integrable over \([a, b]\) ensures that we can apply many integral properties, such as the ones used in our solution. We utilized partitions and upper and lower sums to explore \( G \)'s continuity, illustrating how Riemann integration supports verifying smoothness (continuity and differentiability) of \( G \). This understanding underpins the rigorous foundations of integration ensuring that the sum converges to a real number, maintaining the continuity of \( G \) on \([a, b]\).

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Problem 21

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Problem 23

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Key Concepts

Continuity of Functions

Understanding the continuity of a function is crucial in calculus, as it forms the basis for more advanced concepts such as differentiation and integration. A function is said to be continuous on an interval if small changes in its input result in small changes in its output. Mathematically, a function \( G(x) \) is continuous on the interval \([a, b]\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x-y| < \delta\), it follows that \(|G(x)-G(y)| < \epsilon\).

This concept implies that the function has no abrupt changes or jumps within the interval. In the process of proving the continuity of \( G(x) = \int_{x}^{b} f(t) \, dt \), we showed that the difference \(|G(x) - G(y)|\) can be made arbitrarily small by choosing appropriate partitions, illustrating how integral properties lead to the conclusion that \(G(x)\) is indeed continuous.

Differentiation

Differentiation is the process of finding the derivative of a function, which indicates the rate at which the function's value changes with respect to its input. For the function \( G(x) = \int_{x}^{b} f(t) \, dt \), differentiation involves finding \( G'(x) \), the derivative of \( G \) at a specific point \( x \).

To determine differentiability, especially at a point \( c \) where \( f \) is continuous, we rely on the limit definition of the derivative: \( G'(c) = \lim_{h \to 0} \frac{G(c+h) - G(c)}{h} \). By using properties of \( f(t) \) when it is continuous at \( c \), and through meticulous limit calculations, it's observed that this process leads to \( G'(c) = -f(c) \). This is a fundamental result linking integration and differentiation, often highlighted by the Fundamental Theorem of Calculus.

Mean Value Theorem

The Mean Value Theorem (MVT) for integrals provides a powerful insight that bridges the functions' average value over an interval and its value at some point within the interval. When applied to functions like \( f(t) \), continuous on \([a, b]\), it states there exists a point \( \xi \in (a, b) \) where the integral of the function over \([a, b]\) equals the function's value at \( \xi \) multiplied by the interval's length.

In our exercise, this theorem helps in understanding why \( G \) is differentiable at a certain point. It offers a way to relate the change over an interval back to a single point, facilitating differentiation of \( G \). This relationship is particularly useful when proving \( G'(c) = -f(c) \), confirming the derivative's existence and its relation to function \( f \).

Riemann Integrals

Riemann integrals represent the classical method of defining an integral, focusing on the summation of areas under curves. This approach involves partitioning the domain into small intervals, calculating the sum of function values times small changes in intervals, and taking the limit of these sums as partitioning becomes finer.

For the function \( G(x) = \int_{x}^{b} f(t) \, dt \), verifying that \( f \) is Riemann integrable over \([a, b]\) ensures that we can apply many integral properties, such as the ones used in our solution. We utilized partitions and upper and lower sums to explore \( G \)'s continuity, illustrating how Riemann integration supports verifying smoothness (continuity and differentiability) of \( G \). This understanding underpins the rigorous foundations of integration ensuring that the sum converges to a real number, maintaining the continuity of \( G \) on \([a, b]\).