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f(x) - g(x) is an integer , f(x) is an integer iff g(x) is an integer

Source: 2013 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020
functionalIncreasingInteger

Problem Statement

Let f:R→Rf : R \to R and g:R→Rg : R \to R be strictly increasing linear functions such that f(x)f(x) is an integer if and only if g(x)g(x) is an integer. Prove that f(x)−g(x)f(x) - g(x) is an integer for any x∈Rx \in R.
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