My earlier example was an ad-hoc one that achieved the goal by just having a parameter that one can tweak to make $f$ get closer to (but not touching) the line defined by "$x=y$". One can easily prove that this creates a contractive map that eventually becomes zero because it falls into the inter...

As $x$ increases:   $\sqrt{4x^4-4x^2+2} = \sqrt{(2x^2-1)^2+1}$ decreases if $x \in [0,\frac12\sqrt2]$ and increases if $x \in [\frac12\sqrt2,-\infty)$.   Symmetrically $\sqrt{4x^4-4x^2+2}$ decreases if $x \in (-\infty,-\frac12\sqrt2]$ and increases if $x \in [-\frac12\sqrt2,0]$. Therefore $\sq...