A manager on the brink rarely speaks optimistically of the future but Crystal Palace hired Frank de Boer to be different. He responded to what may yet prove his last game in charge by casting his thoughts forward. A man who displayed great on-field eloquence in his playing days, courtesy of his wonderful left foot, showed another kind in a reaction to a fourth consecutive league defeat. The result seemed to render his own position more precarious. He insisted the display did not.

“I am not disappointed about the performance and it gives me a lot of hope for the future,” the Dutchman insisted, though it remains a moot point if his own is at Selhurst Park. A manager who has failed to beat Huddersfield, Liverpool, Swansea and now Burnley is in a private battle with himself. He lasted 85 days at Internazionale. He is up to 76 at Palace now.

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De Boer’s afternoon at Turf Moor was bookended by displays of schadenfreude, taunts that he would be “sacked in the morning” following both Burnley’s third-minute winner and the final whistle. If so, the newly-hired sporting director, Dougie Freedman, would step in while they looked for an 11th permanent manager since 2010. De Boer himself shrugged off suggestions he had been given an ultimatum: win at Burnley or else. His June appointment was described by the chairman Steve Parish as an “amazing milestone”. He is charged with determining if the Dutchman, four league games into a three-year contract, is now a millstone.

“They have to decide,” De Boer said. “When I’m still the manager of Palace, I will give 100%. I just focus on what I can do. We are still with the project and we know where we come from and what we want to achieve. I’m convinced we are going to achieve it.” He was recruited to implement change and it is undeniable he has had an impact. The Eagles have made their worst start to a season for 92 years and stand alone among the 92 Premier and Football League clubs: everyone else has mustered at least one league goal.

“The only thing we didn’t do was score,” the former Ajax manager rued, his irritation increased by the knowledge Palace had furnished Burnley with the decider. The London club have reached a statistical low in his tenure, but other numbers offered him encouragement: they had 23 attempts at goal. Burnley had four.

“I will sign for this every game if we create that amount of chances against any opponent,” De Boer said. “This is what I want to see from any team I manage. We played with courage. That gives me a lot of hope. It was quite dynamic.”

Play Video 0:42 De Boer: 'I'm focussing on what i can control' –video

That might be stretching it. Palace had 65% of possession but much of it came under the category of sterile domination. Yet it was conceivable that the centre-back Scott Dann could have had a hat-trick. “They had two fantastic chances, especially Dann’s header at the end,” the Burnley manager, Sean Dyche, said.

Dann headed wide, though, and he had been denied by twin goal-line clearances from Matthew Lowton and the excellent James Tarkowski. Christian Benteke had other opportunities. His radar was for once awry. When it functioned correctly, he met unexpected obduracy in the shape of Burnley’s reserve goalkeeper Nick Pope. Yet the damning element for Palace was that they rarely troubled a Premier League debutant.

“Technically, a fantastic save,” Dyche said after the deputy was summoned when Tom Heaton landed awkwardly catching a cross. The England goalkeeper has a suspected dislocated shoulder. “If it is, it is not weeks [out], more like months,” Dyche said. He had “empathy, not sympathy” with De Boer, seeking out his counterpart for a consoling word.

“I thought they were the better side,” he said. “I told him so. They were very good. They played with more freedom.” He attributed that to De Boer’s decision to jettison his favoured 3-4-3 formation for a more familiar 4-3-3.

A winner could afford to be magnanimous. While Palace wrestle with an identity crisis, Burnley know who they are and what they have to do. “We had to fight for every inch,” Dyche said. But a goal came gift-wrapped. Criticised for sideways passing under De Boer, Palace were quick to supply a defence-splitting ball. Sadly for the recalled Lee Chung-yong, the defence he bisected was his own. Brought in to supply Benteke, he instead picked out Chris Wood with a misguided attempt to find the goalkeeper Wayne Hennessey. Burnley’s record signing provided an assured finish to vindicate Dyche, who promoted the £15m forward, the scorer of a late leveller against Tottenham last time out, to the starting 11.

“We gave a very sloppy goal away,” said De Boer. If it suggested his players were eager to hasten his exit, they responded. “We showed spirit,” he said. Yet if Palace are pointless, goalless and possibly luckless and he may soon be jobless, Burnley are buoyant.

They had already beaten Chelsea and drawn with Tottenham. More capital gains capped their profitable start to the season while Palace’s losses are mounting. De Boer’s job may yet be among them.