If f(x) is continuous, I'd say there's no such func. Here's the reason:
1.The simplest case, that f(x) have no zero points in (0,1), so the curve is like bridge. For instance, let f(x)=sin(4*pi*x), and g(x)=f(x)-f(x+2/5), then
g(0)<0, g(3/5)>0, and since g(x) is continuous, there should be...