Definition:Convergent Mapping
Definition
Metric Space
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ except possibly at $c$.
Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.
Then $f$ converges to the limit $L$ as $x$ tends to $c$.
Real Function
As the real number line $\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions:
Let $f: \R \to \R$ be a real function defined everywhere on $A_1$ except possibly at $c$.
Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.
Then $f$ converges to the limit $L$ as $x$ tends to $c$.
Complex Function
As the complex plane $\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions:
Let $f: \C \to \C$ be a complex function defined everywhere on $\C$ except possibly at $c$.
Let $\map f z$ tend to the limit $L$ as $z$ tends to $c$.
Then $f$ converges to the limit $L$ as $z$ tends to $c$.
Also see
- Results about convergent mappings can be found here.