I am confused about antiderivatives of multivariable functions, specifically $\delta(ct-|x|)$ and $\delta(t-|x/c|)$.
Here $\delta(.)$ is the Dirac delta function (distribution)and $x$ and $t$ are variables and $c$ is a constant $>0$.
MY QUESTION IS: What variable(s) are the antiderivative integrals of $\delta(ct-|x|)$ or $\delta(t-|x/c|)$ with respect to, are there separate antiderivatives for each variable (what do they look like), is there such a thing as a 'total antiderivative'?
I know these antiderivatives lead to Heaviside step functions.