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2 problems match “functional inequalities”. Clear

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1. Possible values of f(2018) ●●●○○ 50 ptsLet $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be an even function with $f(1)=1$ and $f(2017) \neq 1$. For all integers $x, y$, $2f(x+y)-f(x)-f(y) \leqslant |f(x)-f(y)|$. Find the value of $f(2018)$.
2. Smallest bound on function differences ●●●○○ 50 ptsThe function $f(x)$ is defined on $[0,1]$ and satisfies $f(0)=f(1)$, and for any $x, y \in [0,1]$ with $x \neq y$, we have $|f(x)-f(y)|<|x-y|$. Find the smallest real number $m$ such that for every such function $f$ and any $x, y \in [0,1]$, we have $|f(x)-f(y)|<m$. Give your answer as a decimal.