Finding a Smooth Function With Given Proprieties I need to find a function $f:\mathbb{R} \to \mathbb{R}$ such that 

$\bullet$ $f$ is smooth ($C^\infty)$

$\bullet$ $|f'(x)| &lt; 1$

$\bullet$ $f(x) \neq x \quad \forall x \in R$

What I've done so far:

I was able to find a function that seems to work (at least looking at the plot). I will show the function bellow:

$$f(x)=
\begin{cases}
e^\frac{-2}{x-2}  &amp; \text{for $x&gt;2$} \\
(x-2)e^{\frac{2}{x-2}} &amp; \text{for $x&lt;2$}\\
0 &amp; \text{for $x=2$}
\end{cases}$$ 

Here is the Plot

But I have 2 problems:

1)I would like to know if there is a simpler function since I would prefer not to define things conditionally

2)I'm not sure I know how to show the function satisfies what I want in a rigorous way.
I'm not sure if it is possible, check https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
I'm not sure if it is possible, check https://en.wikipedia.org/wiki/Fixed_point_(mathematics)