The functional calculus allows us to associate to every function \(f:J\to{\mathbb{R}}\) the operator function \(Of:{\mathcal S}_J({\mathcal H})\to{\mathcal S}({\mathcal H})\), \(A\mapsto f(A)\), where \({\mathcal S}({\mathcal H})\) denotes the real vector space of self-adjoint operators on a finite-dimensional complex Hilbert space and \({\mathcal S}_J({\mathcal H})\) stands for the set of all \(A\in{\mathcal S}({\mathcal H})\) having the spectrum in the interval \(J\subseteq{\mathbb{R}}\). Similarly, if \(J'\subseteq{\mathbb{R}}\) is another interval, \({\mathcal H}'\) is another finite-dimensional complex Hilbert space and \(f:J\times J'\to{\mathbb{R}}\), then one can construct a two-variable operator function \(Of:{\mathcal S}_J({\mathcal H})\times{\mathcal S}_{J'}({\mathcal H}') \to{\mathcal S}({\mathcal H}\otimes{\mathcal H}')\). In its first part, the paper under review is concerned with relations between differentiability properties of \(f\) and \(Of\), and with continuity properties of the map \(f\mapsto Of\) (in both one- and two-variable cases) as well. Several interesting connections between the operator function \(Of\) and its differential are pointed out when the function \(f:J\to{\mathbb{R}}\) is continuously differentiable. In the second part of the paper, the convex cone \({\mathcal OC}_2\) of operator convex functions \(f:(-1,1)\times(-1,1)\to{\mathbb{R}}\) is studied. (A function \(f\) is called operator convex whenever the associated operator function \(Of\) is convex.) A certain three-parameter family of faces of the convex cone \({\mathcal OC}_2\) is investigated.