Definition:Convergent Mapping

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Definition

Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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 $f \left({x}\right)$ 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 $f \left({x}\right)$ 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 $f \left({z}\right)$ tend to the limit $L$ as $z$ tends to $c$.


Then $f$ converges to the limit $L$ as $z$ tends to $c$.


Also see