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Everton boss Marco Silva was left ruing missed chances after feeling his side deserved to leave Crystal Palace with more than a draw.
The Toffees twice hit the woodwork in what was otherwise a drab 0-0 stalemate at Selhurst Park.
Bernard struck the bottom of the post before Lucas Digne's deflected cross clipped the crossbar, while substitute Cenk Tosun saw a clever back-heeled effort brilliantly saved by Palace goalkeeper Vicente Guaita.
After thrashing Manchester United 4-0 last weekend, this was a comedown for Everton – who now sit four points off Wolves in seventh in what could prove to be the final Europa League qualification place.
"We deserved the three points," Silva said after the game.
"It was clear we were the best team on the pitch. We have tried everything to win the match, two balls off the post, a fantastic save from the goalkeeper.
"From the first minute we came here to dominate the game, it is always difficult to do that here, they are strong on the counter-attack.
"We created the chances, being more clinical this afternoon and we win this game clearly.
"We had a good chance for Dominic and in the second half had three big chances to score, we didn't score we tried everything and deserved more.
"Playing like we did this afternoon with this quality, this balance and being solid as we did, that is the most important thing for us."
Palace had their own impressive win last time out as they left Arsenal with a 3-2 victory.
They were largely out-played by Everton, however, and rarely threatened Jordan Pickford's goal, with Eagles boss Roy Hodgson agreeing with Silva that his side were second-best.
"We dreamed of a different game," he said.
"A game where we would have a lot of possession and pass the ball around and show some of the quality we believe we have in the team.
"We weren't able to do that and you could say that is because we were playing against a good team, there are games that don't go the way you dream of going, you have to play the game that is out there to play.
"We accepted it wasn't to be one of our best days on the ball, they were in our final third on a good number of occasions and we had to make sure our defending was as solid and brave as it was.
"Had they scored and won it 1-0 I couldn't have come in here and cried 'scandal' – there were some individuals I thought did better than others but in terms of a team I would say our ability to stay in the game at times, I am satisfied with that."
