Riemann integrability of composite functions

In summary, we are trying to prove that f o g is Riemann integrable on [a,b] if either f or g is a step function. The solution for when g is a step function is known, but the approach for when f is a step function is unknown. The discontinuities of f o g can occur when either g or f is discontinuous, but both sets of discontinuities have measure 0 as f and g are Riemann integrable. It is also noted that this result may differ in the case of Lebesgue integration.
  • #1
darkreignmoon
1
0
Hi, I'm stuck on this problem here about composite function, help is appreciated:

Let g : [a,b] -> [c,d] be Riemann integrable on [c,d] and f : [c,d] -> R is Riemann integrable on [c,d]. Prove that f o g is Riemann integrable on [a,b] if either f or g is a step function

I was able to solve the part where g is a step function but if f is a step function then I don't know how to approach this.
 
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  • #2
Consider the set of discontinuities of fog. f(g(x)) can be discontinuous when either g is discontinuous(say on the set G) or when it attains a 'bad' value in the domain of f(say in F).
Both F,G have measure 0 as f,g are Riemann integrable.FXG has measure 0 if f is a step function & hence the result.
I wonder what happens in case of Lebesgue integration.
 

Related to Riemann integrability of composite functions

1. What is the definition of Riemann integrability for composite functions?

The Riemann integrability of a composite function is the property that allows the function to be integrated over a certain interval using the Riemann integral. It is based on the idea that if the function is continuous and bounded over the interval, then it can be divided into smaller intervals where the function is approximated by a constant value, making it integrable.

2. How is the Riemann integrability of a composite function determined?

The Riemann integrability of a composite function is determined by checking if the function is continuous and bounded over the interval of integration. If the function meets these criteria, then it is considered Riemann integrable and can be evaluated using the Riemann integral.

3. Can a composite function be Riemann integrable if its individual components are not?

Yes, it is possible for a composite function to be Riemann integrable even if its individual components are not. This is because the Riemann integrability of a composite function depends on the overall behavior of the function, not just the behavior of its individual components.

4. How does the Riemann integrability of a composite function relate to the Riemann integrability of its components?

The Riemann integrability of a composite function is closely related to the Riemann integrability of its components. If the components of a composite function are Riemann integrable, then the composite function is also Riemann integrable. However, if the components are not Riemann integrable, the composite function may still be Riemann integrable if it meets the criteria for Riemann integrability on its own.

5. What is the significance of Riemann integrability for composite functions in mathematical analysis?

Riemann integrability for composite functions is significant in mathematical analysis as it allows for the evaluation of complex functions using the Riemann integral. It also provides a way to determine the area under a curve for composite functions, which is a fundamental concept in calculus and many other areas of mathematics.

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