MathNet 2013 Question 20

Let $k$ be a positive integer. Let $f:\mathbb Z\to\mathbb Z$ be a bijection such that for any integers $i,j$ satisfying $|i-j|\le k$, we have $|f(i)-f(j)|\le k$. Prove that $|f(i)-f(j)|=|i-j|$ for all $i,j\in\mathbb Z$.