For simplicity, we will assume in the following that \(A\ \) is included in \(f\) definition, which now become \(\left[m^2s^{-1}\right]\).  The notation used above is congruent with the one used in \cite{Rigon_2016} where, however, more general cases are treated. 
According to the definition of the Heaviside step function the integral giving the age-ranked discharge can be considered for various time steps.
\(q\left(t,t_{in}\right)=0\)
\(q\left(t,t_{in}\right)\ =\ p_{t_{in}}\int_{_{t_{in}}}^tf\left(t-x\right)\ dx\)
because  \(H\left(x\right)H\left(t-x\right)\ \equiv1\)
\(q\left(t,t_{in}\right)\ =\ p_{t_{in}}\int_{_{t_{in}}}^{t_{in}+\Delta t}f\left(t-x\right)\ dx\)
because \(H\left(x\right)H\left(t-x\right)\ \equiv1\) for \(t_{in}<\ x\le t_{in}+\Delta t\ \)  and \(H\left(x\right)H\left(t-x\right)\ \equiv0\ \) for \(t\ >\ t_{in}+\ \Delta t\)
Let's  now define the so called S-function (which, as it is known, correspond to the probability or residence/travel times \cite{Rigon_2016}, in this case multiplied by the catchment area \(A\)