416 Curves, lengths, and curvatures
are, when r > 0, continuous but notpiecewise monotone over 0 5 x < 1. Sup-
posing that p = q = r = 2, show that 0 S f(x) 5 1 and f is differentiable as
well as continuous over 0 <= x << 1. It can be proved that C does not haveY=R2 V_TNI
1finite length.
14 It is worthwhile to cultivate the
ability to understand and even to con-
struct simple examples of curves that
have properties of various sorts. Figure
7.197 shows a part of the graph C of a
g t -i function f, defined over 0 < x :!5; 1, for
Figure 7.197 which f'(x) exists when 0 < x <= 1 and,
moreover, f(x) = f'(x) = 0 when x = 0,
1'1'11,'ff ,'1, g,. We do not bother to give formulas expressing f(x) in terms of
x, being content to describe the graph C. The rectangles have their upper left
corners on the graph of the equation y = x2. The part C1 of the graph in the
rectangle R1 over the interval 5 x 5 1 consists of four monotone parts. Show
that I < IC1I =< 2. We now turn our attention to the rectangle R2 which lies
above the interval 4 < x < and has height (IC) 2. In this rectangle we construct
another part C2 of C, this part consisting of 42 monotone parts so that 1 5 M1- Tell how to continue the construction so that the total resulting curve C
will not have finite length.
15 Let x(t) and y(t) be continuous over a 5 t < b so that the set of points
P(t) for which P(t) = (x(t), y(t)) and a < t < b, so ordered that P(t1) precedes
P(t2) when ti < t2, constitutes a curve C in the xy plane. The curve C is said to
be a closed curve if P(a) = P(b), that is, if the curve ends where it starts. Give
some examples of closed curves and some examples of curves that are not closed.
16 A set A in the xy plane is said to be connected if to each pair P1 and P2
of points in I there corresponds a curve C of the type described in Problem 15
such that P(a) = P1, P(b) = P2, and each point of C is a point of A. Let r
(capital gamma) be the circle with center at Po and radius r. Prove that the
set .41 of points P for which IP0Pl < r (this set being the interior of I') is a con-
nected set. Prove that the set A2 of points P for which I P0PI > r (this being the
exterior of r) is a connected set. Prove finally that if a set A3 contains a point
P1 for which 1P0P0P1I < r and a point P2 for which IPoP2I > r but contains no point
of the circle r, then .4s is not a connected set. Proof of last part: Suppose, intend-
ing to establish a contradiction, that the set A$ is connected. Then there is a
curve C, determined by continuous functions x(t) and y(t) as in Problem 15, such
that P(a) = P1, P(b) = P2, and each point of C is a point of -4a. Let
f(t) _ IPoP(t)I = (xo - x(t))2 + (yo - y(t))
so that f(t) is the distance from Po to P(t). Then f(a) < r and f(b) > r. Since
f is a continuous function, it follows from the intermediate-value theorem
(Theorem 5.48) that there is a number t* for which a < t* < b and f(t*) = T.
The point P(t*) is a point of the circle r, so P(t*) is not in /13 and we have the
required contradiction. Remark: The geometrical nature of a circle in a plane
is so simple that basic facts involving its interior and exterior are easily described
and easily proved. The remaining problems of this list involve more general
situations.