Definition:Convergent Mapping

Convergence of a Function on a 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 $$c$$.

Convergence of Real and Complex Functions
As:
 * The real number line $$\R$$ under the usual metric forms a metric space;
 * The complex plane $$\C$$ under the usual metric forms a metric space;

the definition holds for real and complex functions.

Divergent Function
A function which is not convergent is divergent.