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We have a function $f : [a, b] \rightarrow \mathbb{R}$ and $g(x) = f(x - c)$ on $[a + c, b+c]$. We want to show that $g$ is integrable if $f$ is integrable.

I am able to show that $\int_{a+c}^{b+c} g(x) dx = \int_a^b f(x) dx$ using substitution, but I'm not sure how to say that $g(x)$ is integrable.

Does anyone have any pointers? Thanks!

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consider the two invertible affine maps of the closed unit interval $I$ to the reals: $$ s:x \to a(1-x)+bx \\ t:x \to a(1-x)+bx+c $$ then considered as maps $I \to \mathbb{R}$ we have: $$ f\circ s = g \circ t = \phi $$ $\phi$ is integrable since $f$ is.

but then $\phi\circ t^{-1}= g \circ t \circ t^{-1} = g$ is integrable

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