The line can be so fine, even with all the work Angelique Kerber has done on herself and her game.

And as she hoisted her second Grand Slam singles trophy of the year (and of her career) on Saturday at the United States Open, it seemed appropriate to wind back the tape eight months to the Australian Open.

In the first round there, she faced a match point in the second-set tiebreaker against the unseeded Misaki Doi of Japan and escaped only when Doi’s shot hit the tape and fell back on Doi’s side of the net.

“What would happen had she not won that match point?” Mary Joe Fernandez, the United States Fed Cup captain, asked on Saturday, shortly before Kerber went out and played her latest remarkable match under major pressure to defeat the 10th-seeded Karolina Pliskova, 6-3, 4-6, 6-4, in the Open final.

Lose to Doi in January, and Kerber would not have gone on to win her first major title at the Australian Open, where she beat Serena Williams in a three-set thriller. Lose to Doi in January, and Kerber might never have found the state of mind necessary to experience this remarkable midcareer renaissance at 28.