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?) 
 A: 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.
