The loss to Flamengo, Fluminense's second consecutive and third in four matches, came after a 90th-minute own goal at the Maracana Stadium.
"It's sad because we played well again and had several chances to score," striker Rafael Sobis said. "But things are not going our way. It's tough, football is not fair sometimes. We have to keep fighting, because if we lower our heads now, then it's over."
Some fans protested by invading the team's headquarters early Monday, and local media reported that they threw trash onto one of the team's training fields, destroyed sponsors' billboards and damaged some of the cars in the parking lot, including some belonging to players.
"The fans are sad and so are we," Luxemburgo said. "But it's not over yet, we have to keep working hard to get the results we need. We are still out of relegation zone and we have to make sure that we stay out in the end."
Luxemburgo has been under pressure for quite some time. Local media reported last week that the coach was almost fired after the team's home loss to Vitoria. His dismissal seemed inevitable after the loss to Flamengo on Sunday, but the club again said it was sticking with the coach.
"The team showed improvement and it wouldn't make sense to make a change at this moment," club president Peter Siemsen said.
Luxemburgo, the most successful coach in the Brazilian league, said he is not considering resigning.
"I'm not going to give up," the 61-year-old coach said. "If it's up to me, I'll stay until the end. I'm not going to quit. I'll keep working hard to make sure we are not relegated."
Luxemburgo said three victories in the last five matches could be enough for Fluminense to avoid relegation. The team's next game is at Corinthians next Sunday.
One of the reasons for the team's recent struggles is the absence of Brazil striker Fred, who has been nursing a right muscle injury since September. It remains unclear if he will be available again before the end of the championship.
Fred was part of the Fluminense team that escaped relegation with a remarkable run in 2009, winning six of the last seven matches to barely remain in the top flight. The following year it won the league title, and again in 2012.
Fluminense is not the only traditional Brazilian club fighting to avoid relegation this year. In addition to rival Vasco da Gama, other teams that are in danger this year include Sao Paulo, Flamengo and club world champion Corinthians. Last year, eight-time Brazilian champion Palmeiras was relegated, although it has already secured its return next season.
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