Editing Real analysis (section)

====Riemann integration====
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to tagged partitions of an interval. Let <math>[a,b]</math> be a [[Interval (mathematics)|closed interval]] of the real line; then a '''''tagged partition''''' <math>\cal{P}</math> of <math>[a,b]</math> is a finite sequence

:<math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>

This partitions the interval <math>[a,b]</math> into <math>n</math> sub-intervals <math>[x_{i-1},x_i]</math> indexed by <math>i=1,\ldots, n</math>, each of which is "tagged" with a distinguished point <math>t_i\in[x_{i-1},x_i]</math>. For a function <math>f</math> bounded on <math>[a,b]</math>, we define the '''''Riemann sum''''' of <math>f</math> with respect to  tagged partition <math>\cal{P}</math> as

:<math>\sum_{i=1}^{n} f(t_i) \Delta_i, </math>

where <math>\Delta_i=x_i-x_{i-1}</math> is the width of sub-interval <math>i</math>.  Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The '''''mesh''''' of such a tagged partition is the width of the largest sub-interval formed by the partition, <math display="inline">\|\Delta_i\| = \max_{i=1,\ldots, n}\Delta_i</math>. We say that the '''''Riemann integral''''' of <math>f</math> on <math>[a,b]</math> is <math>S</math> if for any <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that, for any tagged partition <math>\cal{P}</math> with mesh <math>\| \Delta_i \| < \delta</math>, we have

::<math>\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.</math>

This is sometimes denoted <math display="inline">\mathcal{R}\int_{a}^b f=S</math>.  When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) '''''Darboux sum'''''. A function is '''''Darboux integrable''''' if the upper and lower [[Darboux integral|Darboux sums]] can be made to be arbitrarily close to each other for a sufficiently small mesh.  Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.  In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.

The [[fundamental theorem of calculus]] asserts that integration and differentiation are inverse operations in a certain sense.