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Compute g(x,t) with the computer

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September 5, 2010
functionalgebra proposedalgebra

Problem Statement

Let S⊂[0,1]S \subset [0, 1] be a set of 5 points with {0,1}⊂S\{0, 1\} \subset S. The graph of a real function f:[0,1]→[0,1]f : [0, 1] \to [0, 1] is continuous and increasing, and it is linear on every subinterval II in [0,1][0, 1] such that the endpoints but no interior points of II are in SS.
We want to compute, using a computer, the extreme values of g(x,t)=f(x+t)−f(x)f(x)−f(x−t)g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)} for x−t,x+t∈[0,1]x - t, x + t \in [0, 1]. At how many points (x,t)(x, t) is it necessary to compute g(x,t)g(x, t) with the computer?
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