Upper and Lower Riemann Integrals with Examples | Real Analysis

A function is Riemann integrable on [a,b] if it satisfies any of these conditions: (1) Continuous on [a,b]; (2) Monotonic (increasing or decreasing) on [a,b]; (3) Has finitely many discontinuities; (4) Is bounded with countably many discontinuities and finite limit points; (5) Is of bounded variation. The converse is not true - integrable functions need not be continuous or monotonic. Uncountable discontinuities (like the Dirichlet function) always imply non-integrability. For discontinuous functions, Riemann integrability requires boundedness, countable discontinuities, and finite limit points. Functions with infinitely many discontinuities can still be integrable if they accumulate at a finite point.

Several classes of functions are guaranteed to be Riemann integrable: (1) Continuous functions on closed intervals; (2) Monotonic functions (non-decreasing or non-increasing) on closed intervals; (3) Functions with finitely many discontinuities. These conditions provide sufficient criteria for integrability without needing to verify the definition directly.

A function f is Riemann integrable on a closed interval [a, b] if and only if the difference between the upper Darboux sum and the lower Darboux sum approaches zero as the mesh (maximum subinterval length) of the partition approaches zero. This criterion provides a practical test for integrability without computing Riemann sums directly, as it only requires verifying that the oscillation of the function on each subinterval, weighted by subinterval length, becomes arbitrarily small for sufficiently fine partitions.

A bounded function f on [a, b] is Riemann-integrable if and only if the upper integral (supremum of integrals of all step functions greater than or equal to f) equals the lower integral (infimum of integrals of all step functions less than or equal to f). This means that when we take the supremum of all lower sums and the infimum of all upper sums, they converge to the same real number, which becomes the definite integral of f over [a, b].

A bounded function on [a, b] is Riemann integrable if and only if for any ε > 0, there exists a partition where the difference between upper and lower sums is less than ε. This criterion proves integrability of continuous functions and a broad class of discontinuous functions. Functions with infinitely many discontinuities can still be integrable if all discontinuities can be covered by finitely many intervals with arbitrarily small total length. Monotone functions are always integrable on any segment. Piecewise continuous functions (finite discontinuities of first kind) are also integrable. The Lebesgue integral generalizes Riemann integration to handle more complex functions.

The Fundamental Theorem of Calculus states that if f is continuous on [a,b], then the function F(x) = ∫ₐˣ f(t)dt is differentiable and F'(x) = f(x); more generally, if f is Riemann integrable on [a,b] and f(x) = g'(x) for some function g, then ∫ₐᵇ f(x)dx = g(b) - g(a). This theorem connects differentiation and integration, showing that the definite integral of a function equals the difference in values of its antiderivative at the endpoints. The proof uses the Mean Value Theorem and Darboux sums, demonstrating that integrability (not continuity) is the essential condition for this result.

The Fundamental Theorem of Calculus connects Riemann integration with differentiation. Part 1: If f is continuous on [a,b] and F is an antiderivative (F' = f), then ∫ₐᵇf(x)dx = F(b) - F(a). Part 2: If f is differentiable and f' is Riemann integrable, then ∫ₐᵇf'(x)dx = f(b) - f(a). This theorem shows that Riemann integration is equivalent to the anti-derivative concept from high school calculus.

The Riemann integral is defined through upper and lower sums: for any partition, compute lower sum using infima and upper sum using suprema. A function is integrable if the supremum of lower sums equals the infimum of upper sums. Continuity on [a,b] guarantees integrability. The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations: Part 1 shows that the integral function G(x)=∫ₐˣ f(t)dt is differentiable with G’(x)=f(x); Part 2 shows that if F is any antiderivative of f, then ∫ₐᵇ f(x)dx = F(b)-F(a).

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse processes: if f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x) (meaning F'(x) = f(x)), then the definite integral ∫_a^b f(x) dx equals F(b) - F(a). This theorem simplifies integration by allowing us to find antiderivatives and evaluate definite integrals through the difference of function values at the limits, rather than using the original definition of integration.

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse processes. Part 1 states that if f is continuous on [a,b], then d/dx [∫ₐˣ f(t) dt] = f(x), meaning the derivative of the definite integral from a constant lower limit to a variable upper limit x yields the original function f(x). Part 2 states that if F is any antiderivative of f (F' = f), then ∫ₐᵇ f(x) dx = F(b) - F(a). This allows computation of definite integrals without Riemann sums by finding an antiderivative and evaluating at bounds. The theorem connects the geometric interpretation of area with the algebraic process of antidifferentiation.

This section classifies important categories of Riemann integrable functions. The instructor proves that every continuous function on a closed interval is integrable, using the theorem of uniform continuity: for any ε > 0, there exists δ > 0 such that any partition with diameter less than δ makes the difference between upper and lower sums less than ε. The oscillation of a continuous function on each subinterval can be made arbitrarily small. The instructor also proves that every monotone function on a closed interval is integrable, since monotone functions are automatically bounded and their oscillation on each subinterval is bounded by the total variation (difference between endpoint values). These results establish that continuous and monotone functions form important classes of integrable functions.

This section extends the class of integrable functions to include monotone functions and functions with finite discontinuities. First, it proves that monotone functions on closed intervals are integrable: since monotone functions are bounded, and for any ε > 0, one can choose a partition with diameter less than ε divided by the difference between the supremum and infimum of the function, the difference between upper and lower Darboux sums will be less than ε. Second, it proves that functions with only a finite number of points of discontinuity are integrable: for a finite set of discontinuities, one can always construct a finite covering with arbitrarily small total length, satisfying Lebesgue's criterion. This extends integrability beyond continuous functions to include functions with jump discontinuities at finitely many points.

Functions that are guaranteed to be Riemann integrable include: (1) Continuous functions on [a,b], (2) Monotonic functions (either non-decreasing or non-increasing) on [a,b], (3) Functions with a finite number of discontinuities on [a,b]. The Dirichlet function (1 at rational points, 0 at irrational points) is not integrable. The integrability of a function depends on its behavior and the nature of its discontinuities.

Several classes of functions are guaranteed to be Riemann integrable: (1) Continuous functions on [a,b] are integrable because uniform continuity ensures that for any ε > 0, there exists δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε/(b-a), making the difference between upper and lower sums less than ε. (2) Monotonic functions on [a,b] are integrable because for a non-decreasing function, the difference between upper and lower sums equals f(b) - f(a), which can be made arbitrarily small by choosing sufficiently fine partitions. (3) Functions with countable discontinuities are integrable if for every ε > 0, there exists a finite collection of intervals whose total length is less than ε and which cover all points of discontinuity.

This comprehensive section establishes the fundamental classes of Riemann integrable functions. First, continuous functions on closed intervals are integrable, proven using Cantor's uniform continuity theorem and Darboux's criterion. Second, bounded functions with finitely many discontinuities are integrable, proven by enclosing discontinuities in small intervals and using uniform continuity on remaining segments. Third, monotonic functions are integrable, proven using the telescoping nature of oscillation sums. These theorems progressively expand the class of integrable functions from continuous to piecewise continuous to monotonic functions, providing a complete characterization of when Riemann integration applies.

Lebesgue integration, developed by French mathematician Henri Lebesgue, provides a more powerful alternative to Riemann integration by measuring the size of sets in the range (y-axis) rather than partitioning the domain (x-axis); this approach enables integration of highly discontinuous functions like the Dirichlet function (which equals 1 for irrationals and 0 for rationals) by summing the product of function values and the measure of their corresponding sets, and forms the foundation for modern probability theory and expected value calculations.

The Riemann integral uses vertical partitions (intervals on the x-axis) and sums areas of rectangles taken sequentially while preserving the curve's order. The Lebesgue integral groups contributions by their heights (values), first counting all contributions at height y1, then y2, etc. This allows Lebesgue to handle complex bases including unions of intervals and countable sets like the rationals. The Lebesgue integral is less sensitive to discontinuities because it doesn't depend on the order of abscissas. Every properly Riemann-integrable function (continuous, piecewise continuous, regulated) is also Lebesgue-integrable with the same value, but Lebesgue extends to more functions.

Despite its strengths, the Riemann integral has significant limitations: it fails for highly discontinuous functions like the characteristic function of rationals, lacks powerful theorems for interchanging limits and integrals (requiring uniform convergence), and struggles with non-Euclidean spaces. The Lebesgue integral addresses these issues by subdividing the function's range (image) instead of its domain. Rather than approximating with horizontal rectangles, it groups curve portions by function values and approximates with vertical slices of constant height. This approach aggregates contributions of the same height together, creating a more robust integration framework that handles pathological functions and provides superior theoretical foundations.

The Lebesgue Integral extends integration beyond Riemann integration by partitioning the function's range rather than its domain, enabling integration of functions with severe discontinuities like the Dirichlet function; it is built on measure theory, which defines measurable sets through sigma algebras and assigns measures (generalizing length, area, volume) to them, allowing rigorous treatment of limits and convergence via theorems like the Monotone Convergence Theorem and Dominated Convergence Theorem, and forms the mathematical foundation for probability theory where measure spaces correspond to probability spaces.