- #1

- 72

- 0

## Homework Statement

Find formulas for the upper and lower sums of [itex]f[/itex] on [itex]P_n[/itex], and use them to compute the value of [itex]\int_0^1f(x)dx[/itex].

[itex]P_n:=\{\frac{j}{n}:j=0,1,...,n\}[/itex] (a partition of [0,1])

[itex]\[

f(x) = \left\{ \begin{array}{ccc} 0 & 0 \le x < 1/2 \\ 1 & 1/2 \le x \le 1 \end{array} \right. \][/itex]

## Homework Equations

[itex]U(f,P)=\sum\limits_{j=1}^{n} M_j(f)\Delta x_j[/itex] and

[itex]L(f,P)=\sum\limits_{j=1}^{n} M_j(f)\Delta x_j[/itex]

where [itex]M_j=sup f([x_{j-1},x_j])[/itex] and [itex]m_j=inf f([x_{j-1},x_j])[/itex]

if [itex]\lim_{n \rightarrow \infty} L(f,P_n)=\lim_{n \rightarrow \infty} U(f,P_n)[/itex] then this equals [itex]\int_0^1f(x)dx[/itex]

## The Attempt at a Solution

So it is easy to see that this function is bounded on [0,1]. So now we can break this up into the different partitions, but now is where I run into a problem. It is finding the inf and the sup of each interval:

so obviously if both [itex]x_j, x_{j-1}[/itex] are < 1/2 then both inf and sup are 0;

if both [itex]x_j, x_{j-1}[/itex] are >= 1/2 then both inf and sup are 1;

so now it is possible for one case to be [itex]x_j \ge 1/2, x_{j-1} < 1/2[/itex]

in which case sup =1/2 and inf =0.

I am stuck from this point. Any help would be appreciated.