Let \(X\) be a topological space and let \(Y\) be a metric space. A function \(f:X \to Y\) is called graph continuous if the closure of its graph \(G(f)\) contains the graph of a continuous function \(g:X \to Y\). Some sufficient conditions for the continuity of a function \(f:X \to Y\) being the uniform limit of a sequence of graph continuous functions are given. Moreover, it is proven that \(f:X \to R\) is continuous iff it is graph continuous, upper- and lower quasicontinuous.