What Is the Metric Space of the Domain and Range in Complex Integration?

[gotjrgkr]

I've been studying complex analysis. In the part of complex integration, there is no reference about the metric space which a continuous function takes as its domain and range. So, i felt confused.

Before asking my question, i use the definition of continuity as follows
$$\rightarrow$$

According to the text "principles of mathematical analysis" by walter rudin,
continuity is defined as follows :
Let X, Y be metric spaces. Let E be a subset of X.
Then a function f mapping E into Y( that is, f: E ⊆ X → Y ) is said to be continuous at x ∊ E if for every $$\epsilon$$ > 0, there is a $$\delta$$ > 0 such that d$$_{X}$$(x,p)<$$\delta$$ and p $$\in$$ E $$\rightarrow$$ d$$_{Y}$$((f(x),f(p))<$$\epsilon$$.
To define complex integration, we must first define the definite integral of the complex valued function F of real variable on the interval [a,b] ; F : [a,b]$$\subseteq$$X $$\rightarrow$$ C by F(t) = U(t)+iV(t) where C is the set of all complex numbers and U and V are real valued functions of real variable which is continuous on [a,b].

In this expression, Y = C and E = [a,b]. Then what is X? is X the set of the whole complex numbers or a subset {(a,0): a is a real} of C?
and it is written that the functions U and V are real valued functions of real variable which is continuous on [a,b]. what is their metric spaces of the domain and the range of the functions U and V? are they C or {(a,0}: a is real} ?

Let R* = {(a,0): a is real}.
I've learned that for a real valued function U of real variable ( U : [a,b]⊆ R* $$\rightarrow$$ R* ), piecewise continuous can be defined. Is it also possible to define piecewise continuous about the above complex valued function F of real variable?? If so, then what is it?

 

[lippyka]

In answer to your question, X is the set of real numbers R*. The metric space for the domain and range of the functions U and V is also R*. It is possible to define piecewise continuous functions of a complex valued function F of real variable. A piecewise continuous function of a complex valued function F of real variable is a function that is continuous on each part of its domain, but may have discontinuities at certain points in its domain.