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What is the distinction between a bounded function, and a finite function? Is there any example of two functions that satisfies only one of them?

Definition If $|f(x)|<+\infty \forall x \in E$, we say $f$ is finite.

Definition If there exist a finite number $M$ such that $|f(x)| \leq M \forall x \in E$, we say $f$ is bounded.

Thank you.

Community
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1LiterTears
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    Some real analysis texts allow functions to take the value $\infty$. So something like $f(x) = x$ is finite but not bounded, while, say, $f(x) = \frac{1}{\lvert x \rvert}$ for nonzero $x$ and $f(0) = \infty$, would be neither finite nor bounded. – MartianInvader Aug 23 '13 at 22:55
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    With these definitions, every bounded function is finite, but not every finite function is bounded. Consider $$f(x) = \begin{cases} 1/x &, x \neq 0\ 0 &, x = 0\end{cases}$$ – Daniel Fischer Aug 23 '13 at 22:59
  • Oh, got it - so $1/x$ approaches to $\infty$ actually means less than $\infty$, hence is finite. – 1LiterTears Aug 23 '13 at 23:00

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To echo MartianInvader's answer in the comments, $f(x)=x$ is finite but not bounded. That's got to be the simplest possible example!

Chris Culter
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