IMO 1969 - Longlist Problems Question 59

(SWE 2) For each $\lambda$ ($0 \lt \lambda \lt 1$ and $\lambda\ne1/n$ for all $n=1,2,3,\ldots$) construct a continuous function $f$ such that there do not exist $x,y$ with $0 \lt \lambda \lt y=x+\lambda\le1$ for which $f(x)=f(y)$.