Definition:Convergent Function

Definition

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.