156 Chapter^3piecewise regular arc the derivatives from the right and from the left must
be different from zero.
A piecewise smooth arc is also called a contour or a path (mainly in
connection with the theory of integration). These terms are also applied
to any rectifiable arc.
A piecewise continuously differentiable arc (in particular, a piecewisesmooth arc) may be defined as a finite sequence { 11, /2, ... , /n} of contin-
uously differentiable arcs (in particular, of smooth arcs) defined on closedintervals 11 , I 2 , ... , In such that if h = [ak, /h], then
k = 1, ... ,n-1
where 1k: z = zk(t), CY.k :::; t ~ f3k·
If, in addition, zn(f3n) = z 1 (CY. 1 ), we have a piecewise continuously differ-
entiable curve, in particular, a piecewise smooth curve, also called a closed
contour (see Definition 3.24).Definition 3.20 The opposite arc of!= z = z(t), CY. :::; t :::; /3, denoted
-1, is the arc defined by
-1: z = z( CY. + /3 - t),
It is readily seen that the arc -1 has the same graph as 1 but the
opposite orientation. In fact, from CY. :::; t :::; /3 it follows that -/3 :::; -t :::; -a,
and adding a+ /3 to each member of the inequality, we obtain a :::; a+ (3-t :::;
/3, so that t and A = a + /3 -t have the same domain, namely, the interval
[a, /3]: i.e., the mapping A = a+ /3-t, which is obviously one-to-one, maps
the interval [a,/3] onto itself. Hence the ranges of z(t) and z(a + f3 - t),
a:::; t:::; /3, are the same. However, if A1 =a+ /3 -tl and A2 =a+ /3 -t2,
from t1 < i2 it follows that >-2 < >. 1 , and if z 1 = z(>. 1 ), z 2 = z(>. 2 ), then
Z2 -<( Z1.
Alternatively, the opposite arc may be defined by-1= z = z( -t), -/3:::; t:::; -a
Definitiion 3.21 An arc 1 is called simple, and also a Jordan arc, if the
function z = z(t), a:::; t:::; /3, defining the arc is one-to-one, i.e., if z(t 1 ) =
z(t2) is true only for ti = t2 (Fig. 3.18a). If z(t 1 ) = z(t 2 ) for t 1 i-t 2
the graph of the arc crosses itself and it is said to have a double (possibly
multiple) point (Fig. 3.18b). The points of the graph of an arc may all be
double (or multiple) points, as in Example l(b), p. 152.
Definition 3.22 By an open neighborhood of a point z 1 = z(t 1 ), a <
i1 < /3, along the graph of a simple arc/, denoted N 1 (z 1 ), we mean the
image under z = z(t) of some open neighborhood ( t 1 - 8, t 1 + 8) C (a, /3),