9.11. EXERCISES 217

5. In the case where F (x) = x, define a “left sum” as ∑nk=1 f (xk−1)(xk− xk−1) and a

“right sum”, ∑nk=1 f (xk)(xk− xk−1) . Also suppose that all partitions have the prop-

erty that xk − xk−1 equals a constant, (b−a)/n so the points in the partition areequally spaced, and define the integral to be the number these right and left sums getclose to as n gets larger and larger. Show that for f given in 9.6,

∫ x0 f (t) dt = x if

x is rational and∫ x

0 f (t) dt = 0 if x is irrational. It turns out that the correct answershould always equal zero for that function, regardless of whether x is rational. Thisillustrates why this method of defining the integral in terms of left and right sums istotal nonsense. Show that even though this is the case, it makes no difference if f iscontinuous.

6. The function F (x) ≡ ⌊x⌋ gives the greatest integer less than or equal to x. ThusF (1/2) = 0,F (5.67) = 5,F (5) = 5, etc. If F (x) = ⌊x⌋ as just described, find∫ 10

0 xdF. More generally, find∫ n

0 f (x)dF where f is a continuous function.

7. Suppose f is a bounded function on [0,1] and for each ε > 0,∫ ‘1

εf (x)dx exists. Can

you conclude∫ 1

0 f (x)dx exists?

8. A differentiable function f defined on (0,∞) satisfies

f (xy) = f (x)+ f (y) , f ′ (1) = 1.

Find f and sketch its graph.

9. Does there exist a function which has two continuous derivatives but the third deriva-tive fails to exist at any point? If so, give an example. If not, explain why.

10. Suppose f is a continuous function on [a,b] and∫ b

a f 2dF = 0 where F is a strictlyincreasing integrator function. Show that then f (x) = 0 for all x. If F is not strictlyincreasing, is the result still true?

11. Suppose f is a continuous function and∫ b

a f (x)xndx = 0 for n = 0,1,2,3 · · · . Showusing Problem 10 that f (x) = 0 for all x. Hint: You might use the Weierstrass ap-proximation theorem.

12. Here is a function: f (x) =

{x2 sin

(1x2

)if x ̸= 0

0 if x = 0Show this function has a deriva-

tive at every point of R. Does it make any sense to write

∫ 1

0f ′ (x)dx = f (1)− f (0) = f (1)?

Explain.

13. Let f (x) ={

sin( 1

x

)if x ̸= 0

0 if x = 0. Is f Riemann integrable with respect to a continu-

ous integrator on the interval [0,1]? Note that this function is not piecewise continu-ous because the limit from the right at 0 does not exist.