Problem 6 - Basic Analysis: Introduction to Real Analysis

Question

Let \(c \in(a, b)\) and let \(d \in \mathbb{R}\). Define \(f:[a, b] \rightarrow \mathbb{R}\) as $$ f(x):=\left\\{\begin{array}{ll} d & \text { if } x=c \\ 0 & \text { if } x \neq c \end{array}\right. $$ Prove that \(f \in \mathscr{A}[a, b]\) and compute \(\int_{a}^{b} f\) using the definition of the integral (but feel free to use the propositions of this section).

Step-by-Step Solution

Answer

The function is integrable, and \( \int_{a}^{b} f = 0 \).

Step 1: Understand the Properties of the Function f

The function \( f \) is defined on the interval \([a, b]\) such that \( f(x) = d \) when \( x = c \) and \( f(x) = 0 \) for all other \( x eq c \). Essentially, \( f \) is zero everywhere except at the point \( c \), where it takes the value \( d \).

Step 2: Apply the Definition of Integrability

The function \( f \) is known to be piecewise constant with only one discontinuity at \( x = c \). For Riemann integrability, a function must be bounded and have a set of discontinuities of measure zero. Here, \( f \) has a single point of discontinuity, which has measure zero in \( \mathbb{R} \), thus \( f \) is Riemann integrable over \([a, b]\).

Step 3: Compute the Integral of f

To compute \( \int_{a}^{b} f \), consider the fact that the integral essentially sums the areas under the function's curve over the interval. Here, since \( f(x) = 0 \) everywhere except for a single point where \( x = c \), and the width of this point is zero, the integral becomes:\[\int_{a}^{b} f(x) \, dx = \int_{a}^{c} 0 \, dx + \int_{c}^{b} 0 \, dx = 0\] Thus, the integral over the interval, given the properties of the function, results in \( 0 \).

Key Concepts

Piecewise Constant Function

A piecewise constant function is defined as a function that has different constant values over different intervals of its domain. In the exercise, the function \( f \) is a classic example of a piecewise constant function. For every value of \( x \) in the interval \([a, b]\) except \( c \), \( f(x) = 0 \). At \( x = c \), \( f(x) = d \), a non-zero constant. Essentially, the function is constant over each sub-interval except for the point \( c \).
Piecewise constant functions are often easy to handle in mathematics because they change values at a finite number of points. These functions are not continuous everywhere but remain bounded, making them ideal candidates for evaluating specific properties like integrability. When breaking down their definition, such functions appear different in separate segments, yet maintain a consistent jump between intervals.

Discontinuity Measure Zero

The concept of discontinuity measure zero refers to isolating points where a function is not continuous. In more formal terms, these are points where the measure, or 'length', is zero, usually manifesting as distinct points with no area. In terms of Riemann integrability, a function is considered integrable if its set of discontinuities has measure zero.
In the given problem, the function \( f \) has a single point of discontinuity at \( x = c \), where it jumps from 0 to \( d \). However, a single point has a measure of zero in the real number line \( \mathbb{R} \). Thus, even if \( f \) is technically discontinuous at \( c \), this discontinuity does not affect its Riemann integrability because the measure of the discontinuity is zero. Hence, functions like \( f \), with isolated jumps or discontinuities, remain integrable over an interval as long as these discontinuities don't accumulate to form a measurable area.

Riemann Integral Computation

Computing the Riemann integral involves summing the areas under the curve of a function over a specific interval. In simple terms, imagine calculating the area between the x-axis and the function's curve from \(a\) to \(b\).
In this exercise, \( f \) is largely zero across the entire interval until it abruptly takes on the value \( d \) at a solitary point \( x = c \). However, since the width of that single point is zero, and areas calculated the traditional way are defined as "length \( \times \) width," the contribution from that single point does not accumulate any area. Therefore, when calculating\[ \int_{a}^{b} f(x) \, dx \] the integral over segments \([a, c]\) and \([c, b]\) equals zero since \( f(x) = 0 \) everywhere else. Essentially, the concept of Riemann integration over this interval reinforces that the point-wise value at \( x = c \) does not contribute to the integral's value; thus, the integral evaluates to zero.