Let f:[a, b] →ℝ be a bounded function and define g:[-b,-a] →ℝ by g(t):=f(-t) . Show that L(g)=L(

step 1 Okay, hello. So here we want to show that the absolute value of the integral going from A to B o

Let f:[a, b] →ℝ be a bounded function and define g:[-b,-a]...

Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function and define $g:[-b,-a] \rightarrow \mathbb{R}$ by $g(t):=f(-t) .$ Show that $L(g)=L(f)$ and $U(g)=U(f) .$ Deduce that $g$ is integrable on $[-b,-a]$ if and only if $f$ is integrable on $[a, b]$ and in that case the Riemann integral of $g$ is equal to the Riemann integral of $f$. (Compare the proof of part (ii) of Proposition $6.26 .$ )

Key Concepts

Bounded Functions

A bounded function on a closed interval is one whose values do not escape a fixed finite range. This property is essential in Riemann integration because it ensures that the upper and lower Riemann sums are finite and that their limits can be meaningfully compared as the mesh of partitions is refined.

Partitions and Riemann Sums

Partitions involve subdividing the interval into smaller subintervals, and Riemann sums are computed as sums of function values multiplied by subinterval lengths. These approximations form the basis for defining both the lower and upper Riemann integrals, which are the supremum of the lower sums and the infimum of the upper sums, respectively.

Upper and Lower Riemann Integrals

The upper and lower Riemann integrals are defined as the infimum of all upper sums and the supremum of all lower sums over every possible partition of the interval. A function is Riemann integrable when these two quantities coincide. The equality of the upper and lower integrals indicates that the function's area under the curve is well-defined.

Change of Variables (Reflection Transformation)

The reflection transformation, in which a function f on an interval [a, b] is related to a function g defined by g(t) = f(-t) on the interval [-b, -a], is an example of a change of variables. This transformation preserves the structure of the partitions and the sizes of subintervals, thereby maintaining the equality of the corresponding Riemann sums. Such invariance under transformation is a powerful tool in analysis for demonstrating properties like integrability.

Riemann Integrability

Riemann integrability is characterized by the coincidence of the lower and upper Riemann integrals. When a bounded function is integrable, its Riemann sums converge to a unique limit, the Riemann integral. The proof that the reflected function g has the same upper and lower integrals as f demonstrates that the integral remains unchanged under reflections, thereby showing the invariance of the Riemann integral under symmetry transformations.

Transcript

Please give Ace some feedback