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The statement that the simple graph f is bounded on the interval [a, b] means that the X-projection of f includes [a, b] and that there exist horizontal lines α and β such that every point of f whose abscissa is in [a, b] is between α and β.
Suppose that the simple graph f is bounded on [a, b]. The statement that iS is an inner sum for f on [a, b] means that there exists a finite collection D of nonoverlapping intervals filling up [a, b] such that, if the length of each interval in D is multiplied by the largest number which exceeds the ordinate of no point of f whose abscissa is in that interval, then the sum of all the products so formed is iS. The inner sum may be described as based on D and designated as iSD. The statement that oS is an outer sum for f on [a, b] means that there exists a finite collection D of nonoverlapping intervals filling up [a, b] such that, if the length of each interval in D is multiplied by the smallest number which the ordinate of no point of f with abscissa in that interval exceeds, then the sum of all the products so formed is oS or oSD (See Figure 7.1).
Since f is bounded on [a, b], there is a number that no inner sum for f on [a, b] exceeds.
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