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With some effort I can evaluate the integral in closed form: $$I(a)=\int_{1}^\infty (x-\lfloor x\rfloor) x^{-a-1}\,dx=\frac{(1-a)\zeta (a)+a}{(a-1) a},\;\;a>0.$$ Hence the desired equality reduces to a simple consistency equation for $f(a)$ and $g(a)$.