State functions and their arguments
This follows from the first law of thermodynamics; i.e., $$\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V,$$ which identifies $U$ as a function of $S$ and $V$. If in addition we have other forms of work, such as chemical work, then we add a chemical work term $\mu\mathrm{d}N$ to the first law, and introduce an additional dependence of $U$ on $N$.

In addition, we can also replace a given independent variable by its conjugate variable, since, by thermodynamic stability, the independent variable is a monotonic function of its conjugate and vice versa. That is, the following sets of independent variables can all be used to characterize $U$: $$(S, V, N), (T, V, N), (S, P, N), (S, V, \mu), (T, P, N), (T, V, \mu), (S, P, N).$$ Note in particular that $(T, P, \mu)$ cannot be used, because these three variables, all being intensive, are not independent and are related by the Gibbs-Duhem equation.