152 Functions, limits, derivatives
nature of the graph of g. Show that g is discontinuous at each x for which x is
rational and that g is continuous at each x for which x is irrational. Hint: If e is
a given positive number, then the set of numbers x for which g(x) > e contains
only a finite number of elements. This fact is useful. Remark: While interest
in the matter should be postponed, this is an example of a bounded function
having a countably infinite set of discontinuities. Moreover, each subinterval
of the interval 0 < x < 1 contains an infinite set of these discontinuities, but the
set of discontinuities has Lebesgue measure zero. The function g is the famous
corn-popper function.
17 Some people know very much about the function F for which F(r) is the
number of lattice points (points having integer coordinates) lying inside and on
the circle of radius r having its center at the origin. Give at least a little precise
information about F.
18 Give an example of a function f such that 0 < f(x) S 1 when 0 <- x 5 1
and such that f is continuous at each point of the interval 0 < x < 1 except at
19 Give an example of a function which (i) is defined over the closed interval
0 <_ x <= 1, (ii) is continuous over the open interval 0 < x < 1, and (iii) is not
continuous over the closed interval 0 5 x < 1.
20 Show that if xi, x2, x3 and .4, B, C, D, E are constants for which x, < x2 <
x3andC00, D 0,E71- 0, and iff(x) = Ax + B + Clx - xii + DJx - x2l + EIx - xai,then f is continuous and the graph off is a broken line consisting of line segments
joined at vertices whose x coordinates are xl, x2, xa.
21 Let(1) A-) _ -x (x 0)
f(x) = x (0 < x 1)
f(x) =2-x (1 _<_x2)
f(x) = 0 (x 2),so that the graph off is a broken line having corners at the points (0,0), (1,1), and
(2,0). Determine five constants A, B, C, D, E such that(2) f(x) _ Ax + B + CixJ + DJx - 11 + Elx - 21.Hint: For each of the four intervals x S 0, 0 5 x <= 1, 1 5 x <- 2, and x > 2,
replace the left member of (2) by the appropriate expression and replace the right
member of (2) by the appropriate expression not involving absolute-value signs.
Ans.:
f(x) = --x+(xI - Ix-11+ lx-21.3.5 Difference quotients and derivatives Let f be defined over an
interval a < x 5 b and let x be a number for which a < x < b. Let
Ax be a number, which may be positive or negative but not 0, for which
a 5 x + Ax 5 b. We may then set(3.51) y = f(x), y + Ay = f(x + .x),