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Riemann Stieltjes Integration

Suppose that f^2 is Riemann-Stieltjes Integrable. Is f Riemann-Stieltjes Integrable as well? Explain.

Solution Preview

Consider a function y=f(x) where x is in [a, b] as follows (a, b are real numbers):
f(x)=-1 when x is rational; f(x)=1, otherwise.

Clearly, f(x)]^2=1 for all x in [a,b]. In other words, f(x)]^2 is a constant over [a,b]. Hence, f^2 is Riemann Stieltjes Integrable over [a, b].

However, f is NOT Riemann Stieltjes Integrable. Why? ...

Solution Summary

Riemann-Stieltjes integration is investigated.

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