91Ó°ÊÓ

When we talk about piecewise constant functions, we're dealing with functions that are constant over specific intervals. In the given problem, the function \( g(x) \) is defined as piecewise constant because:

• For \( a < x < d \), the function takes the constant value \( g_1 \).
• For \( d < x < b \), it jumps to another constant value \( g_2 \).

This type of function is helpful because it simplifies calculations in integrals due to the constant nature over segments. It only changes values at discrete points, such as \( a \), \( d \), and \( b \) here, making it straightforward to handle when setting up integrals like the Riemann-Stieltjes integral. Each segment is individually simple to integrate because the function doesn't change within those bounds.

Continuity is a critical property of functions addressed in this exercise. A function is continuous if, roughly speaking, there are no sudden jumps or breaks in its values. Here’s how continuity applies to the function \( f(x) \):

• It's continuous from the right at \( a \), which means for values approaching \( a \) from just above, the function doesn't jump suddenly.
• It's continuous from the left at \( b \), ensuring no abrupt changes as we approach \( b \) from below.
• Lastly, the function is continuous at \( d \), meaning it doesn't jump when crossing \( d \).

These continuity conditions make it manageable to compute integrals, like in the Riemann-Stieltjes approach, because they ensure the function behaves nicely over the specified interval, allowing for straightforward limits and integral calculations.

The Riemann-Stieltjes integral generalizes the traditional Riemann integral by integrating one function, \( f(x) \), with respect to another, \( g(x) \). In this context, the integral \[\int_{a}^{b} f(x) d g(x)\]is significant because it incorporates how both functions interact over the interval \([a, b]\). Instead of measuring area just under \( f(x) \), it considers how \( g(x) \) changes.In the exercise, \( g(x) \) alters stepwise at \( d \), marking breaks at specific points, while \( f(x) \)'s continuity ensures no abrupt values around these changes. This integral breaks into two parts:

• \( \int_a^d f(x) d g(x) \) where \( g(x) \) is constant and simplifies due to this constant nature.
• \( \int_d^b f(x) d g(x) \) following a similar simplification.

Ultimately, the Riemann-Stieltjes integral exists due to this structured setup, and combining results from sub-intervals gives us the complete value, clearly connecting how piecewise behavior of \( g(x) \) aligns with the smooth continuity of \( f(x) \).