Real Analysis 53 | Riemann Integral - Properties [dark version]

Riemann integrals come with a set of core “calculus-ready” properties: the integral operator is linear and order-preserving, it behaves predictably when you change the integration interval, and it can be extended to handle reversed limits. These facts matter because they turn integration from a definition into a tool—one that supports algebraic manipulation, interval-splitting, and consistent sign conventions.

For any bounded function f on an interval [a,b], the Riemann integral is defined for those functions that are Riemann integrable. All such functions form a vector space R[a,b], meaning sums and scalar multiples of integrable functions stay integrable. The integral then acts like a well-behaved operator: the map f ↦ ∫ f is linear and monotonic. Linearity means the integral respects addition and scalar multiplication—so ∫(f+g) equals ∫f + ∫g, and ∫(αf) equals α∫f. Monotonicity means order is preserved: if f(x) ≥ g(x) for every x in the interval, then ∫f ≥ ∫g. The justification relies on earlier results for step functions: since Riemann integrable functions can be approximated by step functions, the properties transfer from the step-function case to the general integrable case.

Next come interval properties. If the integral is written with different endpoints, it effectively restricts attention to the region between those endpoints. Formally, ∫[c,d] f is interpreted as the integral of the restricted function f|[c,d], so values of f outside [c,d] don’t matter for that integral. A second key rule is additivity over a split: for any intermediate point c between a and b, the integral over [a,b] equals the sum of integrals over [a,c] and [c,b]. Geometrically, this corresponds to splitting the area under the curve into two parts.

These formulas extend even when f is defined on the whole real line, as long as the actual integration is over a compact interval. But there’s a subtlety when limits are “backwards,” such as when the lower limit exceeds the upper limit. The remedy is a sign convention: reversing the order of integration introduces a minus sign, reflecting the idea that the “orientation” of the interval flips the sign of the accumulated area.

Finally, the transcript highlights which functions are guaranteed to be Riemann integrable. Every continuous function on [a,b] is Riemann integrable, but not every Riemann integrable function must be continuous. As additional sufficient conditions, monotone functions are also Riemann integrable—whether monotonically increasing or monotonically decreasing. The proof of these integrability claims is left for later, but the takeaway is practical: many common functions encountered in analysis are integrable, and the integral’s properties make them workable for computation and theory.