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I was wondering (or mind-wandering) about a function described as:

For any given $x_1$, $x_2$, $x_m = \frac {x_1+x_2} {2}$, $$f(x_m) = f(x_1) + a (f(x_2) - f(x_1)) , a \in ]0;1[$$

For example, $f(x_m)$ is $\frac 23$ of the distance between $f(x_1)$ and $f(x_2)$.

I can intuitively see that such a function isn't consistent if $a \neq \frac 1 2$ but how to prove it mathematically.

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I've removed [differential-equations] and [fractals], which struck me as wishful-thinking tags: you're just sort of hoping something from those fields will be relevant but don't have any concrete reason to expect as much. – Rahul Jun 7 '12 at 7:02
Yeah, well… ahem… thanks. – SteeveDroz Jun 7 '12 at 7:05
up vote 7 down vote accepted

Interchange $x_1$ and $x_2$: $f(x_m) = (1-a) f(x_1) + a f(x_2) = (1-a) f(x_2) + a f(x_1)$, so $(1-2a) (f(x_1) - f(x_2)) = 0$. Thus unless $a = 1/2$, $f(x_1) = f(x_2)$, and $f$ is constant.

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