I want to construct a function $f:[0,1]×[0,1]\rightarrow [0,1]$ such that
- $f(0,t)=t$
- $f(1,t)=2t-1$ $ \forall$ $ t\geq \frac{1}{2}$
- $f(s,t)=0$ $ \forall $ $0 \leq t \leq \frac{s}{2}$
- $f(s,\frac{s}{2})=0$
- $f(s,1)=1$
- $f$ is continuous
I have the function but not sure how it has been constructed rigorously and the procedure for constructing the same
This is the function $f(s,t)=\frac{2t-s}{2-s}$
PS: Also, is such a function unique. If so, how can it be proved ?