Question

Suppose that \(f\) is integrable on \([a, b]\) and define $$ f^{+}(x)=\left\\{\begin{array}{ll} f(x) & \text { if } f(x) \geq 0, \\ 0 & \text { if } f(x)<0, \end{array}\right. \text { and } f^{-}(x)=\left\\{\begin{array}{ll} 0 & \text { if } f(x) \geq 0, \\ f(x) & \text { if } f(x)<0 . \end{array}\right. $$ Show that \(f^{+}\) and \(f^{-}\) are integrable on \([a, b],\) and $$ \int_{a}^{b} f(x) d x=\int_{a}^{b} f^{+}(x) d x+\int_{a}^{b} f^{-}(x) d x . $$

Short answer

Based on the given function \(f(x)\) and its restricted versions \(f^+(x)\) and \(f^-(x)\), we proved that both \(f^+(x)\) and \(f^-(x)\) are integrable on the same interval as \(f(x)\). We then showed that the integral of \(f(x)\) from a to b can be expressed as the sum of the integrals of \(f^+(x)\) and \(f^-(x)\) over the same interval using the linearity of the integral. This result tells us that we can analyze the positive and negative contributions of \(f(x)\) separately when integrating the function over a given interval.

Step-by-step solution

Prove that \(f^+(x)\) and \(f^-(x)\) are integrable

Since \(f\) is integrable on \([a, b]\), it is bounded on this interval. The functions \(f^+(x)\) and \(f^-(x)\) are also bounded on \([a, b]\) since they have the same values as \(f(x)\) when positive and negative, respectively, and have a value of 0 otherwise. Additionally, if \(f(x)\) has a finite number of discontinuities on \([a, b]\), then so do \(f^+(x)\) and \(f^-(x)\) as they are only modified versions of \(f(x)\). Hence, both \(f^+(x)\) and \(f^-(x)\) are integrable on the interval \([a, b]\).

Compute the integral of \(f^+(x)\) and \(f^-(x)\) over the interval \([a,b]\)

We will now show that the integral of \(f(x)\) can be represented as the sum of the integrals of \(f^+(x)\) and \(f^-(x)\). To do this, we first note that \(f(x) = f^+(x) + f^-(x)\) by their definitions. Therefore, the integral of \(f(x)\) over the interval \([a, b]\) can also be expressed as: $$ \int_{a}^{b} f(x) d x = \int_{a}^{b} (f^{+}(x) + f^{-}(x)) d x. $$

Use linearity of the integral to separate the terms

The integral is a linear operator, meaning that it satisfies the following property: $$ \int_{a}^{b} (u(x) + v(x))d x = \int_{a}^{b} u(x)dx + \int_{a}^{b} v(x)dx. $$ Applying this property to the integral of \(f(x)\), we obtain: $$ \int_{a}^{b} f(x) d x = \int_{a}^{b} f^{+}(x) d x+\int_{a}^{b} f^{-}(x) d x. $$ Thus, we have shown that the integral of \(f(x)\) over the interval \([a, b]\) can be expressed as the sum of the integrals of \(f^+(x)\) and \(f^-(x)\) over the same interval, as required.

Key concepts

Integrable Function

An integrable function is a fundamental concept in real analysis. It refers to a function whose integral over a given interval exists. To say that a function \( f \) is integrable on an interval \([a, b]\) means that the function meets certain criteria that allow us to compute its integral.

These criteria usually involve conditions like boundedness and having a limited number of discontinuities. To elaborate:

• **Boundedness:** The function \( f \) should not shoot off to infinity within the interval; it should stay within a range of values.
• **Discontinuities:** While perfect continuity isn't required, the function can't oscillate too wildly. Typically, it's acceptable if there are only a finite number of jumps or breaks (discontinuities) within \([a, b]\).

In our exercise, \( f^+(x) \) and \( f^-(x) \) are modified versions of the function \( f(x) \). Because \( f(x) \) is integrable, both \( f^+(x) \) and \( f^-(x) \) inherit the boundedness and limited discontinuity properties, allowing them to be integrable too.

Definite Integral

The definite integral involves the computation of the integral of a function over a specific interval \([a, b]\). It is a central concept in calculus and analysis, providing a way to accumulate quantities, compute areas, and solve differential equations.

The notation for a definite integral is \( \int_{a}^{b} f(x) \; dx \), representing the area under the curve of \( f(x) \) from point \( a \) to point \( b \). Key features include:

• **Limits of Integration:** These are the lower \( a \) and upper \( b \) bounds, which specify where the accumulation starts and ends.
• **Function Behavior:** The function \( f(x) \) enclosed within the integral sign determines the shape and size of the area we are interested in.
• **Integral as a Sum:** The integral can be thought of as summing up an infinite number of infinitesimally small quantities (areas of rectangles under the curve) over the interval.

In the problem, the integral of \( f(x) \) is shown to be the sum of the integrals of \( f^+(x) \) and \( f^-(x) \). This property leverages how definite integrals represent total or net accumulation - adding the areas above and below the x-axis, inclusive of the transformations \( f^+ \) and \( f^- \).

Linearity of Integration

Linearity is a powerful property of the integral that lets us simplify integrals of sums or scaled functions. This principle is crucial in our exercise to split the integral of \( f(x) \) into parts involving \( f^+(x) \) and \( f^-(x) \):

**Linearity of Integration** states:

• The integral of a sum is the sum of the integrals: \( \int_{a}^{b} (u(x) + v(x)) \, dx = \int_{a}^{b} u(x) \, dx + \int_{a}^{b} v(x) \, dx \).
• Additionally, the integral of a constant times a function is the constant times the integral of the function: \( \int_{a}^{b} c \cdot f(x) \, dx = c \cdot \int_{a}^{b} f(x) \, dx \).

By applying the linearity principle in the context of the exercise, we demonstrate that the integral of \( f(x) = f^+(x) + f^-(x) \) over \([a, b]\) splits nicely into separate, manageable integrals for \( f^+ \) and \( f^- \).

This is particularly advantageous because it breaks down the problem, giving you straightforward parts to compute and interpret.