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 3 of 4: Determine the values on [0,1)[0,1)
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f(0)=0,f(x)≤0 (0≤x<1),f(x)+f(1−x)≥0f(0)=0,\qquad f(x)\le0\ (0\le x<1),\qquad f(x)+f(1-x)\ge0
Detailed analysis

From f(1)=f(0)+1=1f(1)=f(0)+1=1 we get f(0)=0f(0)=0, and condition (ii) gives f(x)≤0f(x)\le0 for 0≤x<10\le x<1. Applying the defining inequality to xx and 1−x1-x gives f(x)+f(1−x)≥0f(x)+f(1-x)\ge0.