Fixed point and non-fixed point function

Question

For constructing another proof I need two functions explicitly and therefore

I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the closed interval $[a,b] \subset \mathbb{R}$ where it is defined) has always somewhere a fixed point for each interval $[a,b]$?

Answer 1

Sure -- $f(x)=x-1$ and $g(x)=x$. I think the latter is the only such function with no restrictions on the interval; for the former, any function which does not intersect $y=x$ works.