Let f:[a,b]-> R be an integrable function.

Show that the graph of f in R^2 has zero content.

(hint: given a partition P of [a,b], interpret Spf-spf as a sum of areas of rectangles that cover the graph of f. )

Not sure where to start, any help?

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- Feb 27th 2011, 05:45 AMcalculuskid1zero content
Let f:[a,b]-> R be an integrable function.

Show that the graph of f in R^2 has zero content.

(hint: given a partition P of [a,b], interpret Spf-spf as a sum of areas of rectangles that cover the graph of f. )

Not sure where to start, any help? - Feb 27th 2011, 06:02 AMFernandoRevilla
:*Hint*

$\displaystyle f$ integrable in $\displaystyle [a,b]$ iff for every $\displaystyle \epsilon>0$ there exists a partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle U(P,f)-L(P,f)<\epsilon$.