MathNet 2010 — Part 02 Question 147

Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbf{R}$ to $\mathbf{R}$ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x) - g(x)$ is an integer.