I'm having a hard time understanding the proof of the following: Let a < b < c and let f: [a,c] -> R be Riemann integrable on [a,b], [b,c] and [a,c]. Then(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx[/tex].

Proof. Let C and C' be the characteristic functions of [a,b] and [b,c] respectively, defined on [a,c]. Then f = C f + C' f and the addition formula above follows from the linearity of the integral.

This is such a facile proof. Sigh. I'm trying to fill in the missing details: I know that since f = C f + C' f, then

[tex]\int_a^c f(x) \, dx = \int_a^c C(x) f(x) \, dx + \int_a^c C'(x) f(x) \, dx[/tex]

where I've used the fact that characteristic functions are Riemann integrable and products of Riemann integrable functions are Riemann integrable. Now how would I show, without much fuss, that

[tex]\int_a^c C(x) f(x) \, dx = \int_a^b f(x) \, dx [/tex]

for example?

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# An Integral Question

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