Why is, in general, an upper (or lower) sum not a special case of
a Riemann sum ? Find a condition for a function f
that the upper and lower sums are actually special cases of
The reason is simple: a Riemann sum requires points where the function
is defined, while an upper/lower sum involves an sup/inf which may not
correspond to points in the range of the function.
What this means is best illustrated via an example. Take the function
Now take the simple partition consisting of the interval [-1, 1]
Since f(x) > 0
for all x
(which is not
visible in the above applet) we must have that
any Riemann sum R(f, P)
must give a value strictly bigger than
. The lower sum with respect to this partition, on the other
hand, is truly zero (which again is not visible in the above
applet (why?)). Therefore this particular lower sum can not be a Riemann
So to find a condition that ensures that upper/lower sums are special cases
of Riemann sums we must ensure that the sup/inf that appears in the definition
of upper/lower sum is a max or min, respectively. In the topology chapter
we have shown that a continuous function over a closed, bounded interval
must have a max and a min.
Therefore, if f is continuous over the interval [a, b]
then the upper and the lower sum are both special cases of a Riemann sum.