Definition:Divergent Function

Definition

A function which is not convergent is divergent.

There are multiple ways that a function can be divergent. Here are some samples:

• Let $f: \R \to \R$ be such that:

$\forall H > 0: \exists \delta > 0: f \left({x}\right) > H$ provided $c < x < c + \delta$

Then (using the language of limits), $f \left({x}\right) \to +\infty$ as $x \to c^+$.

• Let $f: \R \to \R$ be such that:

$f \left({x}\right) = \begin{cases} 0 & : x \in \Q \\ 1 & : x \notin \Q \end{cases}$

Then $x$ converges to neither $0$ nor $1$ and hence is divergent (although, it needs to be noted, not to infinity).

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.16, \ \S 8.20 \ (5)$