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Let $f: [0, 1] \times [0, 1] \to \mathbb{R}$ be defined by

$$f(x, y) = \begin{cases} 0,&\text{if } 0 \le x < \frac{1}{2}\\ 1,&\text{if }\frac{1}{2} \le x \le 1\;. \end{cases}$$

I need to show that this function is integrable, and my instructor says if we consider the partition $P = (P_1, P_2)$ where $P_1 = P_2 = \{0, \frac{1}{2}, 1\}$, then $U(f, P) = L(f,P) = \frac{1}{2}$, but this is clearly false since

$$U(f, P) = (1/2)^2 + (1/2)^2 + (1/2)^2 + (1/2)^2 = 1$$

and

$$ L(f, P) = 0 + 0 + (1/2)^2 + (1/2)^2 = 1/2.$$

Can someone point out where I am going wrong?

  • 0
    We don't even have a partition the way it is defined2012-07-26
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    I think $P$ is a partition of $[0, 1] \times [0, 1]$2012-07-26
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    Jean-Sébastien is right. Specifying a grid is not a description of partition2012-07-26
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    I understand that your partition is supposed to be 4 squares, but you need explcitly and very carefully describe all this 4 squares. Some of them will contain their boundary, others doesn't2012-07-26
  • 0
    I interpret what I see as cutting into four equal squares.2012-07-26
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    But what about bounadries of this squares. It is important for computation of $U(f,P)$ and $L(f,P)$2012-07-26
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    I learned that a partition of a rectangle $[a_1, b_1] \times...\times [a_n, b_n]$ is a collection $P = (P_1,...P_n)$ where $P_i$ is a partition of the interval $[a_i, b_i]$2012-07-26
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    So the subrectangles of my partition will be $[0, \frac{1}{2}] \times [0, \frac{1}{2}], [0, \frac{1}{2}] \times [0, 1], [0, 1] \times [0, 1]$, and $[0, 1] \times [0, \frac{1}{2}]$2012-07-26
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    Take a slightly finer partition that encloses the line $\{\frac{1}{2}\} \times [0,1]$ in a rectangle of area $< \epsilon$. Then it should be straightforward? The only issue is this line, which has content 0.2012-07-26

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