We use the continuum discretized coupled channels method to investigate the effects of continuum-continuum coupling on the breakup and fusion cross sections of the weakly bound 6 Li nucleus with the 58 Ni, 144 Sm and 208 Pb nuclear targets. The cross sections were analyzed at incident energies E cm below, close to and above the Coulomb barrier V B . We found that for the medium and heavy targets, the breakup cross sections are enhanced at energies below the Coulomb barrier (E cm /V B 0.8) owing to these couplings. For the lighter target, relatively small enhancement of the breakup cross sections appear at energies well below the barrier (E cm /V B 0.6). At energies E cm /V B >0.8 for medium and heavy targets, and E cm /V B >0.6 for the light target, the continuum-continuum couplings substantially suppress the breakup cross sections. On the other hand, the fusion cross sections are enhanced at energies E cm /V B <1.4, E cm /V B <1.2 and E cm /V B <0.8 for the light, medium and heavy target, respectively. The enhancement decreases as the target mass increases. Above the indicated respective energies, these couplings suppress the fusion cross sections. We also compared the breakup and fusion cross sections, and found that below the barrier, the breakup cross sections are more dominant regardless of whether continuum-continuum couplings are included.