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Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold?

1. $G(x)\le x$

2. $G(1)<1$

3. $F(x)>0$

4. $\min(y,F(x)) \le G(x+y)-G(x)$

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold?

1. $G(x)\le x$,

2. $G(1)<1$,

3. $F(x)>0$ if $x>0$,

4. $\min(y,F(x)) \le G(x+y)-G(x)$.

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold?

1. $G(x)\le x$

2. $F(x)+G(x)<1$

3. $x>0, G(x)=0 \Longrightarrow F(x)>0$

4. $\min(y,F(x)) \le G(x+y)-G(x)$

If the answer is yes, I'd be more interested in a specific, simpler example, than in a large or complete, but more complex, class of solutions.

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold?

1. $G(x)\le x$

2. $G(1)<1$

3. $F(x)>0$

4. $\min(y,F(x)) \le G(x+y)-G(x)$