Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 1 / 11D / iii

Use the discontinuous exampleIn both pieces $0<f(x)<x$. For $x>1/2$, the image stays strictly above $1/2$, and induction givesThis positive limiting point is not a fixed point, since $f(1/2)=1/4$. The discontinuity is precisely why the limit cannot be passed through $f$. Therefore the pointwise zero-limit conclusion can fail without continuity.