Function which has no fixed points Problem:
Can anyone come up with an explicit function $f \colon \mathbb R \to \mathbb R$ such that $| f(x) - f(y)| < |x-y|$ for all $x,y\in \mathbb R$ and $f$ has no fixed point? 
I could prove that such a function exists like a hyperpolic function which is below the $y=x$ axis and doesn't intersect it. But, I am looking for an explicit function that satisfies that.

For an example that doesn't involve defining the function piecewise, how about
$$f(x) = x \Phi(x) + \Phi'(x)$$
where $\Phi$ is the normal cdf and $\Phi'$ is its derivative, the normal pdf. You can view some of its properties on Wolfram Alpha.
