Pure equilibria when one player wants distance

Two players each choose a real number from $[0,1]$ at the same time. Player 1 receives the absolute difference between the two numbers, and Player 2 pays that same amount (a zero-sum game). How many Nash equilibria in pure strategies does this game have?

$0$. Player 2's only best reply is to copy Player 1's number, giving payoff 0, but whenever the numbers are equal Player 1 can deviate to a farther point and earn something positive. So no pair of pure choices is stable.