function $f:\mathbb{R}\to \mathbb{R}$ of class $C^\infty$ such that $f(x)\not =x, \forall x\in \mathbb{R}$ and $|f^\prime(x)|<1$ Give an example of function $f:\mathbb{R}\to \mathbb{R}$ of class $C^\infty$ such that $f(x)\not =x, \forall x\in \mathbb{R}$ and $|f^\prime(x)|<1, \forall x\in \mathbb{R}$.
The condition of $|f′(x)|<1$ is equivalently a " $f$ is Liphchitziana with constant $c<1$. If g is bounded function of class $C^\infty$ with $|g(x)|<c<1$ we know that function
$f(x)=\int_{0}^x g(t)dt$ is Lipchitziana with constant c<1, of class $C^\infty$ with  $|f′(x)|=|g(x)|<c<1$. So my idea is to find a function $g$ continuous bounded by a number less than 1 so that the function f defined by $f(x)=\int_0^x g(t)dt$ is the one that meets the properties of $|f′(x)|<1$. My problem is in the condition $f(x)\not= x,\forall x\in \mathbb{R}$
 A: Take $f(x)=x-\int_{-\infty}^{x} g(t) \, dt$ where $g$ is any integrable smooth function on $\mathbb R$ such that $0<g(x)<1$ for all $x$. Note that $0<f'(x)<1$. For a specific example take $g(x)=e^{-x^{2}}$. 
A: You said:

The condition of $|f′(x)|<1$ is equivalently a " $f$ is Liphchitziana with constant $c<1$.

That is not correct. The conditions (A)

$$
 |f'(x) | < 1 \text{ for all } x \in \Bbb R
$$

and (B)

There is a number $c < 1$ such that
  $$
|f'(x) | \le c \text{ for all } x \in \Bbb R
$$

are not equivalent.
(B) means that $f$ is Lipschitz continuous with a Lipschitz constant $c < 1$. In that case $f$ necessarily has a fixed point, i.e. $f(x) = x$ for some $x \in \Bbb R$. This can be shown with elementary methods (mean-value theorem and intermediate value theorem), or as as consequence of the Banach fixed-point theorem. 
You said:

My problem is in the condition $f(x)\not= x,\forall x\in \mathbb{R}$

Understandable now, because there is no function with Lipschitz constant less than one, and without fixed points.
But in your case only condition (A) is given, and then an example is
$$
 f(x) = \sqrt{x^2+1}
$$
$f$ is a $C^\infty$ function with $f(x) > x$ and $|f'(x)| < 1$ for all $x \in \Bbb R$.
