Divergent Function/Examples
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Examples of Divergent Functions
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).