\section*{Problem 3 - 101033}

For each real number \(a\), give all linear functions \(f(x)\) that have the property that for every real \(x\), \(f(x)=f(x+1)-a\).
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Let \(f(x)=m x+n\) with real numbers \(m\) and \(n\). Then the condition translates to the statement \(m x+n=\) \(m(x+1)+n-a=m x+n+(m-a)\) or \(m=a\). In fact, all linear functions \(f(x)=a x+n\) with slope \(a\) also satisfy the condition, as can be easily verified by substitution.