The author studies the behavior, under deformations, of normal analytic singularities and their numerical invariants. Let \(\pi: (X,x)\to (C,0)\) be a germ of deformation of normal isolated singularity of relative dimension \(n\geq 2\) with the singular locus S over a one-dimensional parameter space C. We show that the function \(C\to {\mathbb{N}}\) defined by \(\tau \to \sum_{y\in S_{\tau}}\delta_ m(X_{\tau},y)\quad\)is upper semi-continuous for each \(m\in {\mathbb{Z}}\), where \(\delta_ m\) means the pluri-genus, \(\delta_ m=\dim_{{\mathbb{C}}}\Gamma (X-\{x\},{\mathcal O}(mK))/L^{2/m}(X-\{x\})\) with \(L^{2/m}(X-\{x\})\) the set of all \(L^{2/m}\)-integrable m-ple holomorphic n-forms on X-\(\{\) \(x\}\). As one corollary of this theorem, we get that, for a hyperbolic section (H,x) of a normal isolated singularity (Z,x), assuming that (H,x) is again normal isolated, \(\delta_ m(H,x)\geq \delta_ m(Z,x).\) As another corollary of the theorem, we obtain that every small deformation of a 2-dimensional quotient singularity is again a quotient singularity, which was already shown by Esnault-Viehweg. By the way, Steenbrink posed a problem: Is every small deformation of a Du Bois singularity again Du Bois ? The answer is ''yes'' for a deformation of an isolated Gorenstein Du Bois singularity by the key lemma for upper semi-continuity of \(\delta_ m\) and the characterization of an isolated Gorenstein Du Bois singularity. However, without the Gorenstein condition, the answer is no. - In {\S} 4 we give an example of deformation \(\pi: (X,x)\to (C,0)\) of a Du Bois singularity \((X_ 0,x)\) with \(X_{\tau}\) not Du Bois for each \(\tau\in C\), \(\tau\) \(\neq 0\).