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Problem

Give a function $f:\mathbb{R} \rightarrow \mathbb{R}$ , $C^\infty$ such that

$1) |f'(x)|<1$

$2) f(x) \neq x$ for all $x \in \mathbb{R}$

My ideia

The idea is to get a function that tends asymptotically to $y=x$. If you reduce the condition to $|f'(x)|\le 1$, it's easy but I don't know how to proceed with the condition $|f'(x)|<1$.

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  • $\begingroup$ Take the graph of $y=e^{-x}$ and rotate it clockwise about $(0,0)$ thru an angle $3\pi/4.$ $\endgroup$ Mar 21 '20 at 6:29
  • $\begingroup$ $f(x)=\frac{1}{2}x$ also works. $\endgroup$
    – C. Brendel
    Mar 21 '20 at 7:04
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    $\begingroup$ @C.Brendel No, this doesn't work since $f(0)=0$ $\endgroup$ Mar 21 '20 at 7:11
  • $\begingroup$ ahh, you're right $\endgroup$
    – C. Brendel
    Mar 21 '20 at 10:35
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The following function fulfills the requirements (and asymptotically tends to $y=x$):

$$f(x)=x+\frac{1}{1+e^{x}}$$

$\frac{1}{1+e^{x}}$ resembles a sigmoid function, that takes values in range $(0, 1)$ and is monotonically decreasing, i.e. $\frac{d}{dx}\frac{1}{1+e^{x}}=\frac{-e^x}{(e^x+1)^2}<0$, so $\frac{d}{dx}f(x)<1$.

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