Denmark hit the woodwork in the second half and dominated the proceedings but failed to break the deadlock against the tenacious African side.
Against a backdrop of raucous noise provided by the thousands of Tunisian fans in the stands, Issam Jebali had a goal ruled out for offside in the 23rd minute but it was Denmark who provided most of the attacking football in the first half.
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The Danes upped the tempo in the second, with Christian Eriksen firing a rasping left-foot shot that was saved by goalkeeper Aymen Dahmen in the 69th minute, and Andreas Cornelius hit the post with a header from the resulting corner.
Tunisia defended resolutely and threatened on the counter as the Danes struggled to break them down. In the end their fans were delighted with a point that set them up for their next game against Australia, while Denmark face France in their second group game.
(With inputs from agencies)