# Simple continuous game or not?

Two players simultaneously pick a number from [0, 1]. Payoff of the first player (equal to the loss of the second) is the distance between those numbers.

1. Does there exist pure-strategy Nash equilibrium?
2. Does there exist mixed-strategy Nash equilibrium? How many?

I found value of the game v=1/2, pure optimal strategy for the second player y=1/2 and the mixed optimal strategy for the first player x={0, 1} with probabilities p=(1/2, 1/2).

I am a bit stuck with finding other mixed strategies, could you help me, please?

1. There is no pure-strategy Nash equilibrium: Assume player 1 choose $$a\in[0,1]$$ and player 2 choose $$b\in[0,1]$$, show that either player 1 or player 2 can do better.