Examples 7.1.6(c):

Find conditions for a function so that the upper sum can be computed by always taking the left endpoint of each subinterval of the partition, or conditions for always being able to take the right endpoints.
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To ensure, for example, that the left endpoint will always be used for the computation of a lower sum we must ensure that regardless of the partition that was chosen the function takes its smallest value inside every partition subinterval on its left endpoint.

We will leave it to you to find the correct condition(s), but you can use the applet below to experiment with various functions.
Click on Options and use the functions:
  • f(x) = x2
  • f(x) = 1 - x2
For one of them the right sum is identical to the upper sum, for the other it is identical to the lower sum. Perhaps that helps finding the right conditions.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007