New classes of discontinuous functions narrower than the first Baire class are introduced and studied. Let (X,T(X)) and (Y,T(Y)) be topological spaces. A mapping \(R: 2^ X\to 2^ Y\) is called a strict regularizer of \(f: X\to Y\) if for each \(x\in X\) and each \(L\in T(Y)\), f(x)\(\in L\), there exists \(G\in T(X)\), \(x\in G\), such that for each \(A\in 2^ X\), \(x\in A\subset G\) implies f(x)\(\in R(A)\subset L\). A function f for which there exists at least one strict regularizer is said to be strictly regularizable. The set of all strictly regularizable mappings f is denoted by SR(X,Y). A mapping \(f: X\to Y\) is said to be precontinuous if there exists an increasing sequence \(\{F_ n\}\), \(F_ n\subset X\), of closed subsets of X such that \(\cup^{\infty}_{n=1}F_ n=X\) and each restriction \(f| F_ n\) is continuous. The class of precontinuous mappings of X into Y is denoted by UC(X,Y). A mapping \(f: X\to Y\) is said to be simple if for any open set L in Y the inverse image \(f^{-1}(L)\) is simultaneously an \(E_{\sigma}\)-set and a \(G_{\delta}\)-set. The class of simple mappings f is denoted by S(X,Y). One of the author’s statements asserts that if X is a complete metric space and Y is a regular topological space, then \(S(X,Y)=UC(X,Y)=SR(X,Y)\).