We introduce a continuous weight attack strategy and numerically investigate the effect of continuous use a weight coefficient ω to define the attack intensity. The weight coefficient ω increases continuously from 1 to infinity, where 1 represents no attack and infinity represents complete destructive attack. Our results show that the continuous weight attack on two selected nodes with small ω (ω≈ 3) could achieve the same damage of complete elimination of a single selected node on both BA and ER networks. It is found that the continuous weight attack on a single selected edge with small ω (ω≈ 2) can reach the same effect of complete elimination of a single edge on BA network, but on ER network the damage of the continuous weight attack on a single edge is close to but always smaller than that of complete elimination of edge even if ω is very large.