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I have two players $A$ and $B$, the action of $A$ is $x_A\geq 0$ and the action of $B$ is $x_B\geq 0$. Let $c_0\in(0,1)$, $c_3>0$ and $c_2>c_1>0$ be constants. The payoff functions of $A$ and $B$ are:

$$U_A(x_A;x_B)=\begin{cases} 1-c_3x_A,~&\mbox{if }c_0>\frac{c_1+x_B}{c_2+x_A+x_B},\\ 1/2-c_3x_A,~&\mbox{if }c_0=\frac{c_1+x_A}{c_2+x_A+x_B},\\ 0-c_3x_A,~&\mbox{otherwise}. \end{cases}$$

$$U_B(x_B;x_A)=\begin{cases} 0-c_3x_B,~&\mbox{if }c_0>\frac{c_1+x_B}{c_2+x_A+x_B},\\ 1/2-c_3x_B,~&\mbox{if }c_0=\frac{c_1+x_A}{c_2+x_A+x_B},\\ 1-c_3x_B,~&\mbox{otherwise}. \end{cases}$$

So basically, $A$ and $B$ are competing on two fractions, $c_0$ and $\frac{c_1+x_A}{c_2+x_A+x_B}$. If $c_0>\frac{c_1+x_A}{c_2+x_A+x_B}$, then $A$ gets everything and $B$ gets nothing (they also need to pay the cost associated with their action), otherwise, it is the other way round.

I am not sure where to start to analyze this game. I was thinking of taking the first order condition, but it seems not to work for this game. I think that this game will have mixed strategy over the continuous set. But I only the textbook example with discrete action space. Any suggestions are appreciated!

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There are papers analyzing games with discontinuous payoffs. One example is https://econtheory.org/ojs/index.php/te/article/viewFile/20170079/17038/503 (and the references therein). Maybe your game satisfies the conditions of one of these papers.

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