Riemann Integrabe Step Functions

In summary, a Riemann Integrable Step Function is a type of function that can be integrated using the Riemann integral. It is defined by a sequence of intervals where the function is constant within each interval and has a finite number of discontinuities. It is different from other types of functions because it is piecewise constant, meaning it is only defined on a finite number of intervals with a constant value in each interval. The Riemann Integral of a Riemann Integrable Step Function is the limit of a sum of rectangles that approximate the area under the curve of the function. To determine if a Step Function is Riemann Integrable, one must check if it is bounded and has a finite number of discontinuities
  • #1
Arkuski
40
0
Suppose that [itex]g:[a,b]\rightarrow[c,d][/itex] is Riemann integrable on [itex][a,b][/itex] and [itex]f:[c,d]\rightarrow ℝ[/itex] is Riemann integrable on [itex][c,d][/itex]. Prove that [itex]f\circ g[/itex] is Riemann integrable on [itex][a,b][/itex] if either [itex]f[/itex] or [itex]g[/itex] is a step function.

The proof for [itex]g[/itex] being a step function seems easy enough, but the other way seems much trickier. Thoughts?
 
Physics news on Phys.org
  • #2
Have you tried doing a formal composition:

ciχAi °g

And then extending by linearity to

Ʃk=1nckχAkog ?
 

What is a Riemann Integrable Step Function?

A Riemann Integrable Step Function is a type of function that can be integrated using the Riemann integral. It is defined by a sequence of intervals where the function is constant within each interval and has a finite number of discontinuities.

How is a Riemann Integrable Step Function different from other types of functions?

Riemann Integrable Step Functions are different from other functions because they are piecewise constant, meaning they are only defined on a finite number of intervals and have a constant value within each interval. Other functions, such as polynomials or trigonometric functions, may have infinitely many points and a varying value at each point.

What is the Riemann Integral of a Riemann Integrable Step Function?

The Riemann Integral of a Riemann Integrable Step Function is the limit of a sum of rectangles that approximate the area under the curve of the function. This integral is used to calculate the area under the curve of the function over a given interval.

How do you determine if a Step Function is Riemann Integrable?

A Step Function is Riemann Integrable if it is bounded and has a finite number of discontinuities within a given interval. This can be determined by calculating the upper and lower Riemann sums and checking if they converge to the same value.

Why are Riemann Integrable Step Functions important in mathematics?

Riemann Integrable Step Functions are important in mathematics because they provide a way to approximate the area under a curve, which is a fundamental concept in calculus. They also serve as a building block for more complex functions and integrals, and are used in various applications such as physics, engineering, and economics.

Similar threads

Prove limit comparison test for Integrals
    • Calculus and Beyond Homework Help
Replies
2
Views
829
Volume of a solid of rotation, obtained rotating a function around x=2
    • Calculus and Beyond Homework Help
Replies
2
Views
858
Linearly independent functions with identically zero Wronskian
    • Calculus and Beyond Homework Help
Replies
1
Views
252
Proving that a "composition" is harmonic
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Compute the residue of a function
    • Calculus and Beyond Homework Help
Replies
2
Views
854
Universal Mapping Property of Free Groups: Definition and Proof
    • Calculus and Beyond Homework Help
Replies
4
Views
907
The integral of a function ##f(x)## from its graph
    • Calculus and Beyond Homework Help
Replies
15
Views
1K
I Original definition of Riemann Integral and Darboux Sums
    • Topology and Analysis
Replies
14
Views
2K
Which of the following statements are true? (Real Analysis question)
    • Calculus and Beyond Homework Help
Replies
15
Views
1K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
568

Hot Threads

Recent Insights

Back
Top