7.4 Basic Existence and Additivity Theorems 391Definition 7.4.11 A bounded function f: [a, b] --+JR is said to be piecewise
continuous on [a, b] if there is a partition P = { x 0 , XI, x 2 , · · · , Xn} of [a, b]
such that V 1 :::; i :::; n, f is continuous on (xi-I, xi)· Notice that one-sided
continuity of f at the partition points Xi is not required by this definition.
Similarly, a bounded function f : [a, b] --+ JR is said to be piecewise mono-
tone on [a,b] ifthere is a partition P = {x 0 ,xI,x 2 , · · · ,xn} of [a,b] such that
V 1 :::; i :::; n, f is monotone on (xi-I, xi)·Definition 7.4.12 A function T : [a, b] --+ JR is said to be a step function
if there is a partition P = {xo,XI,x2, · · · ,xn} of [a,b] and 3 real numbers
CI, c2, · · · , Cn such that V 1 :::; i :::; n,
T(x) =Ci if Xi-I < X <Xi·
That is , a step function is constant on the interior of each subinterval
created by consecutive points of the partition P. We could have called a step
function a "piecewise constant" function. Notice that the values of T(xi) for
Xi E P are completely unconstrained by this definition.Theorem 7.4.13 (Bounded) piecewise continuous functions, piecewise mono-
tone functions, and step functions relative to a partition P = { x 0 , XI, x 2 , · · · , Xn}
of [a, b] are all integrable on [a, b]. Their integrals obey the formulaProof. Exercise 8. •Using the concept of step functions we can gain greater geometric insight
into the nature of Riemann integrability of functions. Step functions allow us
to formulate a geometrically appealing condition equivalent to integrability.Theorem 7.4.14 (Step Function Squeeze Criterion for Integrability) A bounded
function f: [a, b] --+JR is integrable on [a, b] if and only if Ve> 0, 3 step Junc-
tions CJ, T relative to some partition P of [a, b] such that
(a) Vx E [a, b], CJ(x) :::; f(x) :::; T(x);(b) J:(T - CJ)<€. (See Figure 7. 7.)[In words, a bounded function is integrable on [a, b] if and only if it can be
squeezed between two step functions that enclose an arbitrarily small area.]