Complex Analysis - Maths KU

122 Chapter 2 Complex Functions and Mappings
In Definition 2.9 we defined continuityof a complex function f at a single
point z0 in the complex plane. We are often also interested in the continuity
of a function on a set of points in the complex plane. A complex function
f is continuous on a set S if f is continuous at z0 for each z0 in S. For
example, using Theorem 2.2 and the limits in (15) and (16), as in Example
5, we can show that f(z) =z2 − iz + 2 is continuous at anypoint z0 in the
complex plane. Therefore, we saythat f is continuous on C. The function
f(z) =
1
z2 is continuous on the set consisting of all complex z such that
+1
z �= ±i.
Properties of Continuous Functions Because the concept of
continuityis defined using the complex limit, various properties of complex
limits can be translated into statements about continuity. Consider Theorem
2.1, which describes the connection between the complex limit of f(z) =
u(x, y)+iv(x, y) and the real limits of u and v. Using the following definition
of continuityfor real functions F (x, y), we can restate this theorem about
limits as a theorem about continuity.
Continuity of a Real Function F (x, y)
A function F is continuous at a point (x0, y0) if
lim
(x,y)→(x0,y0) F (x, y) =F (x0, y0). (20)
Again, this definition of continuityis analogous to (17). From (20) and Theorem
2.1, we obtain the following result.
Theorem 2.3 Real and Imaginary Parts of a Continuous Function
Suppose that f(z) =u(x, y) +iv(x, y) and z0 = x0 + iy0. Then the
complex function f is continuous at the point z0 if and onlyif both real
functions u and v are continuous at the point (x0, y0).
Proof Assume that the complex function f is continuous at z0. Then from
Definition 2.9 we have:
lim f(z) =f(z0) =u(x0, y0)+iv(x0, y0). (21)
z→z0
ByTheorem 2.1, this implies that:
lim
(x,y)→(x0,y0) u(x, y) =u(x0, y0) and lim
(x,y)→(x0,y0) v(x, y) =v(x0, y0). (22)
Therefore, from (20), both u and v are continuous at (x0, y0). Conversely, if
u and v are continuous at (x0, y0), then
lim
(x,y)→(x0,y0) u(x, y) =u(x0,y0) and lim
(x,y)→(x0,y0) v(x, y) =v(x0, y0).