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14 1. Measure theory integrable, in which case the Riemann integral and Darboux integrals are equal. Exercise 1.1.23. Show that any continuous function f : [a, b] → R is Rie- mann integrable. More generally, show that any bounded, piecewise contin- uous8 function f : [a, b] → R is Riemann integrable. Now we connect the Riemann integral to Jordan measure in two ways. First, we connect the Riemann integral to one-dimensional Jordan measure: Exercise 1.1.24 (Basic properties of the Riemann integral). Let [a, b] be an interval, and let f, g : [a, b] → R be Riemann integrable. Establish the following statements: (1) (Linearity) For any real number c, cf and f + g are Riemann inte- grable, with b a cf(x) dx = c · b a f(x) dx and b a f(x) + g(x) dx = b a f(x) dx + b a g(x) dx. (2) (Monotonicity) If f ≤g pointwise (i.e. f(x) ≤ g(x) for all x ∈ [a, b]), then b a f(x) dx ≤ b a g(x) dx. (3) (Indicator) If E is a Jordan measurable of [a, b], then the indica- tor function 1E : [a, b] → R (defined by setting 1E(x) := 1 when x ∈ E and 1E(x) := 0 otherwise) is Riemann integrable, and b a 1E(x) dx = m(E). Finally, show that these properties uniquely define the Riemann integral, in the sense that the functional f → b a f(x) dx is the only map from the space of Riemann integrable functions on [a, b] to R which obeys all three of the above properties. Next, we connect the integral to two-dimensional Jordan measure: Exercise 1.1.25 (Area interpretation of the Riemann integral). Let [a, b] be an interval, and let f : [a, b] → R be a bounded function. Show that f is Riemann integrable if and only if the sets E+ := {(x, t) : x ∈ [a, b] 0 ≤ t ≤ f(x)} and E− := {(x, t) : x ∈ [a, b] f(x) ≤ t ≤ 0} are both Jordan measurable in R2, in which case one has b a f(x) dx = m2(E+) − m2(E−), where m2 denotes two-dimensional Jordan measure. (Hint: First establish this in the case when f is non-negative.) 8A function f : [a, b] → R is piecewise continuous if one can partition [a, b] into finitely many intervals, such that f is continuous on each interval.