f(x) = x except at the end points where it equals (a+b)/2AxiomOfChoice said:Can someone give me an example of a bounded function [itex]f[/itex] defined on a closed interval [itex][a,b][/itex] such that [itex]f[/itex] does not attain its sup (or inf) on this interval? Obviously, [itex]f[/itex] cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous, bounded function for which [itex]\sup\limits_{x\in [a,b]} f(x) \neq \max\limits_{x\in [a,b]} f(x)[/itex].
This is much easier than my example. Thanks!lavinia said:f(x) = x except at the end points where it equals (a+b)/2
A Riemann sum is a method of approximating the area under a curve by dividing the region into smaller rectangles and adding up their areas.
The Riemann sum is used to find the area under a curve by approximating the area of the region with smaller and smaller rectangles. As the width of the rectangles approaches zero, the sum of their areas approaches the exact area under the curve.
The sup (supremum) and inf (infimum) are used in Riemann sums to determine the upper and lower bounds of the rectangles. The sup represents the maximum value of the function within a given interval, while the inf represents the minimum value. These values are used to calculate the height of the rectangles in the Riemann sum formula.
Riemann sums are closely related to definite integrals, as they are both methods of finding the area under a curve. As the width of the rectangles in a Riemann sum approach zero, it becomes a definite integral, which is the exact area under the curve.
Yes, Riemann sums can be used to approximate the area under any curve as long as the function is bounded and continuous within the given interval. However, as the width of the rectangles decreases, the accuracy of the approximation increases.