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...Spivak s calculus on manifolds solutions manual thomas hughes august chapter functions euclidean space p prove that x n i proof if e is the usual basis rn then we can write and thus when does equality hold in theorem notice get y so have precisely are dependent sgn xi for all one a non negative multiple of other conditions same as similarly combining these results yields which implies desired quantity called distance between interpret geometrically triangle inequality z let f g be integrable rb b holds must some r what continuous c show special case way to this would observe xf t k by integrability sums considered tagged partition such each will approach respectively continuity square root function absolute value taking limit willgive us result however following hint either there exists rsuchthat or since nonnegative from lebesgue riemann rst almost everywhere therefore bf bg...