# Real-Valued Function With Distinct Average Rates of Change

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?)

• 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. Commented Sep 19, 2019 at 21:14
• @CecilEllard I highly suspect that no such continuous function exist, but that some such discontinuous function exists. Commented Sep 19, 2019 at 21:15

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.

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.