MathLabs

Problem 1

Let f:R→Rf:\mathbb R\to\mathbb R satisfy f(x)+f(y)+1≥f(x+y)≥f(x)+f(y)f(x)+f(y)+1\ge f(x+y)\ge f(x)+f(y) for all real x,yx,y; f(0)≥f(x)f(0)\ge f(x) for x∈[0,1)x\in[0,1); and −f(−1)=f(1)=1-f(-1)=f(1)=1. Find all such functions.
Step 1 of 4: Apply the inequality with y=1y=1
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f(x)+1≤f(x+1)≤f(x)+2f(x)+1\le f(x+1)\le f(x)+2
Detailed analysis

Using f(1)=1f(1)=1 in the defining inequality gives f(x)+1≤f(x+1)≤f(x)+2f(x)+1\le f(x+1)\le f(x)+2.