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Let $f, g:\Bbb R \to \Bbb R$ be bounded functions satisfying $$ |f(x+y)-f(x)g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb R$.

Prove or disprove $$ |f(x)||1-g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb R$.

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