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contintue to do sth.是做完一件事继续做另一件事的意思，continue doing sth.是继续做刚才没做完的那件事。
continue to do和doing区别
侧重点不同
1.continue to do
解析：continue一般后面常接不定式，比较普遍，而很少用现在分词。不过在作及物动词时，把现在分词看做动名词也是可以的。
2.continue doing
解析：continue用作及物动词时可接名词、动名词、动词不定式或that从句作宾语,也可以接as短语充当补足语的复合宾语。continue还可引出直接引语。
含义不同
1.continue to do
释义：继续做某事，继续做事，继续做另一件事。
2.continue doing
释义：一直做（同一件事）。
continue to do和doing例句
After he finished reading a novel,he continued to play games with his friends.
他读完小说后跟朋友们继续玩游戏.
We hope to continue to have her close support and friendship.
我们希望她能继续大力支持，并能与她维持亲密友谊。
I hope they continue to fight for equal justice after I'm gone
我希望我走了以后他们能继续为司法公正而战。