Walter Mazzarri's side dominated possession for most of the early stages, but struggled to create chances as Udinese's stubborn defence frustrated the hosts.
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Hernanes was the next player to be frustrated for Inter as he watched a long-range free kick drift just inches wide of the target.
The Brazilian came even closer to breaking the deadlock just a few minutes later when his powerful strike from the edge of the box forced Scuffet to tip the ball past the post.
Inter continued to look the more likely of the two sides to take the lead until the final whistle, but Udinese held firm to claim a point