for any finite partition P = {[x.sub.1],[x.sub.2] ,..., [x.sub.k] ,..., [x.sub.n]and arbitrarily chosen [[epsilon].sub.i] [member of] [[x.sub.i]1,[x.sub.i](1 [less than or equal to] i [less than or equal to] n) whenever [lambda] (P) < [delta]([epsilon], [lambda]).Further I is called the

Riemann integral of f in the ([epsilon], [lambda]) topology over [a, b] , denoted by [[integral].sup.b.sub.a] f(t)dt.

The above inequalities have been used in [4] and [5] for obtaining inequalities between special means and on estimating the error in approximating the Stieltjes integral [[integral].sub.a.sup.b] f (x) du (x) in terms of the

Riemann integral for the function f and the divided difference of u.

But this is just the Riemann-Lebesgue lemma for

Riemann integrals [31].

Dubois and Prade [3] generalized the

Riemann integral over a closed interval to fuzzy mappings.

The book covers the set of real numbers, elementary point-set topology, sequences and series of real numbers, limits and continuity, differentiation, the

Riemann integral, sequences and series of functions, functions of several real variables, the Lebesgue integral, and many other related subjects over nine chapters.

In a calculus textbook (see [1] or [2]), the

Riemann integral [[integral].sup.b.sub.a] f(x)dx for a continuous function f over the interval [a, b] represents the net area of the region bounded by the curve y = f(x), two vertical lines x = a and x = b (both perpendicular to the x-axis).

Using examples and very useful illustrations she describes the basic building blocks of real analysis, including sets and set notations, functions, and sequences, the real numbers, measuring distances, sets and limits, continuity, real-valued functions, completeness, compactness, connectedness, differentiation of functions of one real variable, iteration and the contraction mapping theorem, the

Riemann integral, sequences of functions, differentiation of functions, truth and probability, number properties, exponents, sequences in R, limits of functions from R to R, doubly indexed sequences, sub-sequences and convergence, series of real numbers, probing the definition of the Reimann integral, power series, and Newton's method.