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England was the better team for long periods at Al Bayt Stadium but ultimately fell short with Olivier Giroud's 78th-minute header proving to be decisive as Les Bleus clinched a 2-1 win to set up a semi-final with Morocco on Thursday (AEDT).
Southgate's contract runs until the end of the European Championship in 2024 but he will use the next few weeks to reflect on whether he wants to carry on.
"Whenever I have finished these tournaments I have needed time to make the correct decision because you go through so many feelings," he said.
"I want to make the right decision, whatever that is for the team, for England and for the Football Association.
"It's right to take some time to do that because I know how my feelings have fluctuated in the immediate aftermath of previous tournaments.
"It's the decision to go again, it's a lot of energy and you have to make sure you're ready for that.
"There are Euro qualifiers in March and there's too much in my head to think about that tonight.
"I wanted to focus on this tournament and I think we've given a really good account of ourselves. But in the end only one team wins and we've fallen just short."
7 - This is the seventh time England have been eliminated from the quarter-final stages of the World Cup, more than any other side in the history of the competition. Heartbreak. pic.twitter.com/T8na9NtD6T— OptaJoe (@OptaJoe) December 10, 2022
Southgate led England to the semi-finals of the World Cup in Russia four years ago and the final of Euro 2020 last year, yet his critics have continued to accuse him of playing in a negative manner.
That certainly wasn't the case in Qatar, with no team scoring more than England's nine goals in the group phase before a 3-0 round of 16 win over Senegal.
And against France, England played front-foot football that troubled the defending champion throughout.
"I think it's the best we've played against a major nation in the period I have been in charge but the scoreline is all that matters," Southgate said.
"They know how close they've come. They know they've pushed a top nation all the way.
"We had more possession, more attempts on goal… I am very proud of how they've been, not just tonight but throughout the tournament.
"We've shown a different face in terms of how we've played.
"We wanted to go toe to toe with the top nations, that's how we approached this tournament and we've done that.
"There's so much to be excited about given the age of some of the players.
"We've shown the rest of the world that English football is healthy and we have some very good players, not just for now but for the future as well."
England captain Harry Kane had the chance to take the game to extra time after Giroud's goal but blazed an 84th-minute penalty over the crossbar.
He had earlier netted from the spot to cancel out Aurelien Tchouameni's first-half opener and Southgate was quick to defend the Tottenham striker.
Keep your head held high, skipper.— England (@England) December 10, 2022
We're proud of you ❤️ pic.twitter.com/oEVyhcvWZ5
"He's very low but he's got nothing to reproach himself for," he said.
"We're in the position we are as a team due to his goals and his leadership over a long period of time. The result was due to 100 minutes of football and things that happened at both ends of the pitch.
"Even if the penalty goes in we've still got a lot to do to win the game. We win and lose together.
"It was difficult when you get a second penalty against a goalkeeper you know very well [Kane's Tottenham team-mate Hugo Lloris]. Even the best are going to miss at times, that's football.
"But there's no one I'd rather have in that situation and if we got a penalty tomorrow I'd feel the same way.
"It's cruel for him because he will feel disappointed in himself, but he shouldn't."