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Let R be the set of all real numbers. Define the average rate of change of a function on the interval [a,b] in the usual way: (f(b)-f(a))/(b-a). Does there exist a real-valued function defined on all of R such that for any two distinct intervals [a,b] and [c,d], the average rate of change on [a,b] does not equal the average rate of change on [c,d]? (In other words, does there exist a real-valued function defined on all of R whose average rates of change are all distinct?)

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    $\begingroup$ Can $f$ be any function, or does it need to be continuous? My instinct in either case is no, but I feel surer about continuous functions. $\endgroup$ Commented Sep 19, 2019 at 21:14
  • $\begingroup$ @CecilEllard I highly suspect that no such continuous function exist, but that some such discontinuous function exists. $\endgroup$ Commented Sep 19, 2019 at 21:15

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There is no such continuous function. There is such a discontinuous function.

The statement is equivalent to saying that the function does not have two parallel chords. In the figure below the heavy line is our function. Draw a chord $AB$ from $(A,f(A))$ to $(B,f(B))$. Let $E$ be the point on the curve furthest from $AB$, which is there because the distance is a continuous function on a compact interval, so it attains its maximum. Find $C$ between $A$ and $E$ with $(C,f(C))$ on the same side of $AB$ as $(E,f(E))$. A line through $C$ parallel to $AB$ will intersect the curve at $(D,f(D))$. The average rate of change over $[A,B]$ and $[C,D]$ is the same.

enter image description here

If the function is allowed to be discontinuous, we can construct one as follows: well order the reals in type $\mathfrak c$ so each has $\lt \mathfrak c$ many predecessors. Call them $r_0,r_1,r_2,\ldots$. Start with $r_0,r_1$ and pick any $f(r_0),f(r_1)$ you want. Now for each point $r_k$, find all the slopes of the lines between all pairs of points that have been chosen so far. There are $\lt \mathfrak c$ of them. Now construct all the lines from all the points with all the slopes and note where they cross $x=r_k$. There are still $\lt \mathfrak c$ points accounted for, so we can find an $f(r_k)$ that does not duplicate any of the existing slopes. Keep going until you have accounted for all the reals.

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