Can the court continue proceedings in the absence of counsel for either party?

I will urge you to look at the relevant rules of court. The rules gives Courts the discretion to strike out the matter or adjourn it in the absence of the parties. However, where the court adjourns the matter, the partly affected can apply to the court within the stipulated time to re-enlist the matter in the causelist.

Yes the court can. This is what the court of appeal

In the case of OMABOE V.KWAME had to say. In that case a party refused to make her case because her counsel was absent in court. The court entered judgment against that party who appealed the decision to set aside the judgment.

The question for decision is whether the ends of justice require us to hold that in the instant case, the court should assume inherent power to set aside the judgment appealed from. As we said, that judgment was obtained in the presence of the party who seeks to have it set aside. He was afforded an opportunity to meet his opponent's case by such evidence as he had. He declined to do so. No blame of any sort can be attributed to the respondent or indeed the court. Ideally, judgment should only be given when the court has heard all parties to a case. But no party is obliged to lead evidence, much less the defendant. There is justice to the plaintiff as well as the defendant and it would be unjust to the plaintiff if the court were to give itself power to vacate a judgment regularly obtained by the plaintiff in accordance with its own rules when the other party, while present at the hearing, chose, for reasons of his own, not to meet his opponent's case. It can hardly be the plaintiff’s fault that the defendant's lawyer was absent from court. Accordingly, like the judge below, we are also of opinion that this is not a case in which the court can exercise its inherent jurisdiction to set aside the judgment appealed from. We note that the appellant's lawyer wrote to the court to apprise it of the reason for his absence. That fact may well absolve counsel from any charge of contempt or discourtesy to the court but aside of that, it contributes nothing to the resolution of the question whether the court has inherent power or that the purposes of justice require it to take upon itself power to vacate the judgment obtained in the circumstances of this case.

I hope this helps.

Yes