The phrase "assertions continued" is correct and usable in written English.
It can be used in contexts where you are discussing ongoing claims or statements that are being made or reiterated.
Example: "In the meeting, the speaker's assertions continued to emphasize the importance of transparency in the project."
Alternatives: "claims persisted" or "statements carried on".
The statement continued: "His assertions were totally without merit or foundation and he then decided not to take the matter further.
The company vowed to stay in business but despite Martosella's assertions Crazy Eddie continued to falter.
One refrain was that the evidence was not settled, an assertion that continued as the industry itself was beginning to warn drivers about driving while phoning.
Boehner's office immediately rejected that assertion and continued to insist that the White House still hasn't gotten serious about spending cuts.
Baroness Thatcher is here suffering from dementia, but this involves only picturesque symptoms: gentle, heartbreaking hallucinations about her late husband, vivid flashbacks to her early underdog struggle, defiant assertions of continuing feistiness with doctors and younger politicians.
Microsoft's continued assertions of innocence were one reason for his ruling, he said.
It was a show of support for steelworkers who had lost their jobs, an act of resilience and an assertion of continuing shared identity.
The continued assertion that Mr Scappaticci is in fact Stakeknife has resulted in very serious risk to his life".
And their assertion that deflation continued during the recovery is even farther from the truth than their description of what happened to industrial production.
The continued assertion by the Bush administration that things are going well in Iraq, and further, that Defense Secretary Donald H. Rumsfeld should be commended on a job well done is quite a fabrication, as was the weapons-of-mass-destruction story.
