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Let say $ a $ a is some constant. I need to find function $f(x)$ that meets this criteria:
$$ \lim_{x\to \infty }(f(x)) = a $$ $$ \lim_{x\to -\infty }(f(x)) = -a $$ $$ f(0) = 0 $$

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3 Answers 3

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Functions

$$x\mapsto \frac {2a}\pi \arctan x$$

$$x\mapsto \begin{cases} \frac x{1+x} & \text{for } x\ge 0 \\ \frac x{1-x} & \text{for } x\le 0\end{cases}$$

seem to fit your criteria.

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  • $\begingroup$ Thanks man.. The first one was the one I needed $\endgroup$ Commented Jul 21, 2017 at 10:58
  • $\begingroup$ @MaxAdamyan The other one has additional property: its value is rational iff its argument is rational. However, it's not that smooth as the first one. $\endgroup$
    – CiaPan
    Commented Jul 21, 2017 at 11:00
  • $\begingroup$ This is a very good function. Not just continuous, in fact infinitely differentiable. $\endgroup$
    – TRUSKI
    Commented Jul 21, 2017 at 11:01
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If continuity is not required:

$f(x)=a$ for $x>0$, $f(0)=0$ and $f(x)=-a$ for $x<0$

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Here is another example: \begin{align*} x\mapsto\frac{ax}{\sqrt{x^2+1}} \end{align*}

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