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Prove that $$\int_a^b f(x)\,dx = c\int_{a/c}^{b/c} f(cy)\,dy$$

I've tried to use subtitution rule, but I'm afraid I can't do this if f is not continuous.

Thanks.

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2  
you can use substitution. It is legal here. –  Sabyasachi Mar 15 at 14:36
    
can you explain please why it's legal here? –  user135245 Mar 15 at 14:39
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The substitution $y=x/c$ is differentiable. –  Unwisdom Mar 15 at 14:41
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@Unwisdom has explained it. –  Sabyasachi Mar 15 at 14:42

2 Answers 2

Substituting $x=g\left(y\right)$ is allowed here under certain conditions on function $g$ (so not conditions on the integrand $f$). Here you do that with function $g\left(y\right)=cy$ which is okay if $c\neq0$.

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This really answers the question. The asker forgot to mention that $c \neq 0$, and you brought this up. Yes. This is certainly true. –  NasuSama Mar 15 at 14:45

Let $x=cy$. Therefore, $dx=c dy$, and the limits become the desired ones: $a/c$ and $b/c$. Therefore, the integral is:

$$\int_a^b f(x)\,dx = c\int_{a/c}^{b/c} f(cy)\,dy$$

Which is what we wanted.

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does not answer the question. –  Sabyasachi Mar 15 at 14:43
    
@Sabyasachi. Why ? This is correctly proven as long as $c$ is not zero. –  Claude Leibovici Mar 15 at 14:50
    
@ClaudeLeibovici the OP did consider the substitution. (S)he isn't sure if it is correct. That explanation is required. –  Sabyasachi Mar 15 at 15:03
    
The OP's confusion regarding the requirement on the continuity of $f$(which is wrong) should also have been cleared up. –  Sabyasachi Mar 15 at 15:04

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