The Riemann integral is a definition of the definite integral that assigns an accumulated value to a function by taking limits of sums over increasingly fine subdivisions of an interval. For nonnegative functions, it represents the area beneath the graph; for functions taking both signs, it represents signed area. Named after Bernhard Riemann, who formulated it in his 1854 habilitation thesis, it is a foundational construction in calculus and mathematical analysis. (math.ucdavis.edu)
Definition through partitions and sums
Let (f:[a,b]\to\mathbb R) be bounded, with (a<b). A partition of the interval is a finite sequence [ P:\quad a=x_0<x_1<\cdots<x_n=b. ] Choose a sample point, or tag, (\xi_i\in[x_{i-1},x_i]) in each subinterval. The corresponding Riemann sum is [ S(f,P,\xi)=\sum_{i=1}^{n}f(\xi_i)(x_i-x_{i-1}). ] The mesh of the partition is [ |P|=\max_i(x_i-x_{i-1}), ] the width of its largest subinterval. (math.ucdavis.edu)
The function is Riemann integrable if a real number (I) exists such that, for every (\varepsilon>0), some (\delta>0) satisfies [ |S(f,P,\xi)-I|<\varepsilon ] for every partition with (|P|<\delta) and every choice of tags. This common limit is written [ I=\int_a^b f(x),dx. ] The requirement concerning every sufficiently fine partition and tag choice is essential: convergence along one specially selected sequence of sums does not establish integrability. Equal-width subintervals are permitted but not required. (math.ucdavis.edu)
Upper and lower sums
An equivalent formulation, called the Darboux integral, avoids choosing tags. On each subinterval define [ m_i=\inf_{[x_{i-1},x_i]}f,\qquad M_i=\sup_{[x_{i-1},x_i]}f. ] The lower and upper sums are [ L(f,P)=\sum_i m_i\Delta x_i,\qquad U(f,P)=\sum_i M_i\Delta x_i. ] Every tagged sum lies between them. Adding partition points can only increase the lower sum and decrease the upper sum. Integrability is equivalent to the existence, for each (\varepsilon>0), of a partition for which [ U(f,P)-L(f,P)<\varepsilon. ] Thus, integrability means that the discrepancy between upper and lower approximations can be made arbitrarily small. Equivalently, the supremum of all lower sums equals the infimum of all upper sums. This formulation is especially useful in proofs. (math.ucdavis.edu)
Integrability and discontinuities
Every continuous function on a closed, bounded interval is Riemann integrable. A standard proof uses uniform continuity to control variation within sufficiently short subintervals. Every real-valued monotonic function on such an interval is also integrable, as is every bounded function with only finitely many discontinuities. Continuity everywhere is therefore sufficient, but not necessary. (math.ucdavis.edu)
The precise characterization is Lebesgue’s criterion: a bounded function on ([a,b]) is Riemann integrable exactly when its set of discontinuities is a null set for Lebesgue measure. Equivalently, it must be continuous almost everywhere. A zero-measure set can be covered by countably many intervals whose total length is arbitrarily small. Every countable set has measure zero, although some uncountable sets do as well. (ocw.mit.edu)
A basic counterexample is the indicator function of the rational numbers on ([0,1]): [ D(x)= \begin{cases} 1,&x\in\mathbb Q,\ 0,&x\notin\mathbb Q. \end{cases} ] Because rational and irrational numbers each form a dense set, every nondegenerate subinterval contains both values. Consequently, every lower sum is zero and every upper sum is one. The function is discontinuous everywhere and is not Riemann integrable. This example also shows why a countable set of nonzero values need not imply Riemann integrability. (math.mit.edu)
Basic properties and evaluation
For integrable (f) and (g), integration is linear: [ \int_a^b(\alpha f+\beta g),dx =\alpha\int_a^b f,dx+\beta\int_a^b g,dx. ] It preserves inequalities and is additive across adjacent intervals. Products and absolute values of integrable functions are integrable, and [ \left|\int_a^b f(x),dx\right| \leq\int_a^b|f(x)|,dx. ] Reversing the integration bounds changes the sign; integration over an interval of zero length gives zero. (math.ucdavis.edu)
The fundamental theorem of calculus connects integration with the derivative. If (f) is continuous and [ F(x)=\int_a^x f(t),dt, ] then (F'(x)=f(x)) at interior points. Hence, if (A) is an antiderivative of (f), [ \int_a^b f(x),dx=A(b)-A(a). ] For example, (\int_0^1x^2,dx=1/3), obtained from (A(x)=x^3/3). The theorem supplies an evaluation method; the integral itself is defined independently of whether a convenient antiderivative is available. (math.ucdavis.edu)
Relation to Lebesgue integration
Every Riemann-integrable function on a bounded interval is Lebesgue integrable, and the two integrals agree. The converse fails: (D) above has Lebesgue integral zero because its nonzero values occur on a countable set. Lebesgue integration, developed within measure theory, accommodates broader classes of measurable functions rather than requiring upper and lower interval approximations to coincide. (math.mit.edu)
The proper Riemann integral requires boundedness on a finite interval. Unbounded functions and infinite intervals are instead handled by an improper integral, defined through additional limits of proper integrals. For example, [ \int_0^1x^{-1/2},dx :=\lim_{\varepsilon\downarrow0} \int_\varepsilon^1x^{-1/2},dx=2. ] This convergence does not make the integrand properly Riemann integrable on ([0,1]): it remains unbounded near zero. (math.ucdavis.edu)