Functions. Limits and Continuity. Arcs and Curves 157y y0 x 0 x
(a) (b)
Fig. 3.186 > O, of the point ti, i.e.,N.y(zi) = {z: z = z(t), a< ti - 6 < t <ti+ 6 < /3, 6 > O}
In a similar way, the following types of neighborhoods may also be
defined:N 1 (zi) = {z: z = z(t), a< ti - 6 ~ t ~ti + 6 < /3, 6 > O}
N 1 (zo) = {z: z = z(t), a~ t <a+ 6 < /3, 6 > O}
N 1 (z*) = {z: z = z(t),a < /3-6 < t ~ /3,6 > O}
Definition 3.23 A function J(z) whose domain of definition contains the
graph of the arc I is called contin71,ous along I if the composite function
f(z(t)) is a continuous function oft on the closed interval [a, /3].
Definition 3.24 An oriented arc is called an oriented closed curve (or a
loop) if its endpoints coincide, i.e., if z(a) = z(/3).
It follows from this definition that a closed curve is not a simple arc
since it has at least a double point. However, the following definition is
in current usage.
Definition 3.25 An oriented simple closed curve, or a Jordan curve, is a
closed curve whose only double point is the point z( a) = z(/3). In other
terms, a closed curve is said to be simple if z(ti) = z(t2) with ti < t 2
implies that ti = a and t 2 = /3.
An oriented simple closed curve may also be defined as a closed curve
that is homeomorphic to the oriented unit circle Co: z = e it, 0 ~ t ~ 27f
(which is the simplest example of a Jordan curve).
The graph of a closed curve is defined as for arcs. Also, the terms
differentiable, continuously differentiable, smooth, piecewise smooth, and so
on, are defined as for arcs.