- #1

dRic2

Gold Member

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Take this function:

##f(x) =

\begin{cases}

0, & x \lt 1 \\

1, & x = 1 \\

0, & x > 1

\end{cases}

##

According to Spivak's Calculus I, a function is integrable on ##[a, b]## (let's take ##[0, 2]## here) if ##inf \{ U(f, P) \} = sup \{ U(f, P) \} ##. Where ##P## is a partition of ##[a, b]## and ##U## means Upper areas, ##L## lower areas.

So, skipping some math, this means I have to show ##U(f, P) - L(f, P) < ε## for every ##ε > 0##.

It actually easy to show and I don't need an explanation on the math, but my problem is that I can't imagine it. I mean, let's try to draw Upper and Lower areas of this function. First wi divide ##[a, b]## into a finite number of parts and then we draw rectangles who approximate the function. It is clear that Lowe areas are always zero, so also upper areas has to be zero. But this means the partition of ##[a, b]## should exclude the point ##x = 1##, but if I exclude ##x = 1## then I'm not integrating over ##[0, 2]## but on ##[0, 1)## and ##(1, 2]##.

Sorry for all the "buts"...

Can someone help me this. I really can't imagine it