We establish one-body reduced density-matrix functional theory for the
canonical ensemble in a finite basis set at elevated temperature. Including
temperature guarantees differentiability of the universal functional by
occupying all states and additionally not fully occupying the states in a
fermionic system. We use convexity of the universal functional and
invertibility of the potential-to-1RDM map to show that the subgradient
contains only one element which is equivalent to differentiability. This allows
us to show that all 1RDMs with a purely fractional occupation number spectrum
($0 < n_i < 1 \; \forall_i$) are uniquely $v$-representable up to a constant.
