A Hausdorff space is SQ-closed if its continuous image in any Hausdorff space is sequentially closed. A mapping \(f: X\to Y\), where \(X\) and \(Y\) are Hausdorff spaces, is weakly continuous if, for each \(x\) in \(X\) and open neighborhood \(V\) of \(f(x)\), \(\bar V\) contains \(f(U)\) for some neighborhood \(U\) of \(x\); also, \(f\) has strongly closed graph if, for \(x\) in \(X\) and \(y\) in \(Y\) with \(f(x)\neq y\), there are openneighborhoods \(U\) of \(x\) and \(V\) of \(y\) with \(f(U)\) and \(\bar V\) disjoint. The author obtains the following new characterization of SQ-closed spaces: if \(S\) is a class of first countable spaces containing all first countablecompletely normal and fully normal spaces, then \(y\) is SQ-closed if and only if, for each \(X\) in \(S\), each mapping of \(X\) into \(Y\) with strongly closed graph isweakly continuous. In fact, \(Y\) is SQ-closed if and only if each mapping of the one-point compactification of the positive integers into \(Y\) with strongly closed graph is weakly continuous.