# Definition:Divergent Function

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*This page is about divergent functions. For other uses, see Divergent (Analysis).*

## Definition

A function which is not convergent is **divergent**.

## Examples

### Function Tending to $+\infty$

Let $f: \R \to \R$ be such that:

- $\forall H > 0: \exists \delta > 0: \map f x > H$ provided $c < x < c + \delta$

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

That is, $f$ is divergent at $c$.

### Function with Values for Rational Numbers

Let $f: \R \to \R$ be such that:

- $\map f x = \begin {cases} 0 & : x \in \Q \\ 1 & : x \notin \Q \end {cases}$

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

## Sources

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