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Are affine transformations on $\mathbb{R}$ the only transformations that preserve ratios of intervals?

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Hint: by following your transformation $T$ by a translation, we can assume $T(0) = 0$. By following that by a scaling, we can assume $T(1) = 1$ (unless $T$ is constant). Now if $T$ preserves ratios of intervals, what must $T(x)$ be?

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    Question: Is it only affine transformations that have this property; or is again going to be that only amongst the continuous functions do affine transformations have this property, while some discontinuous functions also do?2012-12-02