Let \(M^n\) be the space of \(n\times n\) complex matrices, and let \[ \lambda :M^n\rightarrow \mathbb{C}^n,\;\lambda (X)=(\lambda _1(X),\dots ,\lambda_n(X)) \] be the eigenvalue map, where \(\lambda_1(X),\dots ,\lambda_n(X)\) denote (repeated according to multiplicity) the eigenvalues of \(X\) ordered lexicographically: if \(k\operatorname{Re}\lambda_l(X)\), or \(\operatorname{Re}\lambda_k(X)=\operatorname{Re}\lambda_l(X)\) with \(\operatorname{Im}\lambda_k(X)\geq \operatorname{Im}\lambda_l(X)\). Any function \(f\circ \lambda :M^n\rightarrow [-\infty ,\infty]\) corresponding to a function invariant under permutation of its argument components \(f:\mathbb{C}^n\rightarrow [-\infty ,\infty]\) is called a spectral function (or also, an eigenvalue function). The spectral function \(f \circ \lambda \) corresponding to a continuous function \(f:\mathbb{C}^n\rightarrow [-\infty ,\infty]\) is a function continuous on \(M^n\) regarded as Euclidean space with the real inner product defined by \(\langle X,Y \rangle = \operatorname{Re}\sum_{r,s}\bar{x}_{rs}y_{rs}\). The spectral abscissa \(\alpha :M^n\rightarrow [-\infty ,\infty]\), \(\alpha (X)=\max\operatorname{Re}\lambda_r(X)\) and the spectral radius \(\rho :M^n\rightarrow [-\infty ,\infty]\), \(\rho (X)=\max|\lambda_r(X)|\) are continuous spectral functions, but they are neither convex nor Lipschitz on \(M^n\). The variational properties of spectral functions are analysed by using the notion of subgradient, extensively presented by \textit{R. T. Rockafellar} and \textit{R. J.-B. Wets} [Variational analysis, Springer-Verlag, New York (1998; Zbl 0888.49001)]. In the last part of the article, the authors introduce the notion of semistable program (maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive) and derive a necessary condition for a local maximizer of a semistable program.