| No.
Consider the limit of the following process.
Define L(0) = a1, H(0) = a2 M(0) = (L(0) + H(0))/2
Now f(M(0)) is either > (F(L(0)) + F(H(0))/2 or its not.
If it is greater, then define L(1) = M(0), H(1) = H(0),
else define L(1) = L(0), H(1) = M(0)
and repeat (ie, its basically a bisection approach choosing the
flatter side of the two sides each time).
Now, what can you say about LIM(M(i))?
You can say that the area �0.5**j*(a2-a1) about it is bounded by
�0.5**(j-1)*(b2-b1)
So the function is continuous at that point.
/Bevin
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| reply .0
A monotone function can only have jump discontinuities, i.e. the
left hand limit at a point of discontinuity is less than the right
hand limit at that point (if the function is monotone increasing,
reverse if not). Since each jump discontinuity defines an interval
and each interval contains a rational (rationals are dense in the
reals) it follows that there can only be a countable number of such
discontinuities. This show that the answer to your question is
no.
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