The continuous coagulation-fragmentation equation with diffusion describes the space and time evolution of the size distribution function \(u\geq 0\) of a population of particles which may either increase their sizes by binary coagulation or decrease their sizes by fragmentation, and move in space according to Brownian movements. The range of the admissible sizes is an open interval \((0,y_0)\) with \(y_0\in (0,\infty]\), and an additional mechanism, the so-called volume scattering, is taken into account if \(y_0n/2\), (ii) the nonnegativity of \(u\) if \(u_0\) is initially nonnegative and (iii) the conservation of mass, that is, \[ \int_\Omega \int_0^{y_0} u(t,x,y)\, dy\,dx = \int_\Omega \int_0^{y_0} u_0(x,y)\, dy\,dx, \] for \(t\) ranging in the maximal existence time. The approach used here differs from that used in previous papers by the authors where the equation was rather viewed as a vector-valued parabolic semilinear equation in \(L^p(\Omega,L^1(0,y_0))\). Global existence results are also provided under suitable assumptions on the reaction terms in the following cases: (i) \(n\leq 3\) and the diffusion coefficient \(d\) does not depend on \(y\), (ii) \(n=1\), (iii) \(L_b\equiv 0\) and \(u^0\) is small enough. In particular, the last result relies on the conservation of mass, the nonlinearity of the reaction term and the fast decay of the linear part of the equation in the space of functions with vanishing mean value.