For metric spaces \((X,d_X)\) and \((Y,d_Y)\) let \(G\) be the space of all continuous functions from a closed subset of \(X\) to \(Y\). The following convergences in \(G\) are considered: Kuratowski convergence, \(\tau \)-convergence, Kuratowski convergence on compacta \(\tau ^c_K\), and Hausdorff convergence on compacta \(\tau ^c_H\). The definitions of these convergences are introduced in this paper and the equivalence of the following statements is proved: (a) \(X\) is locally compact, (b) the Kuratowski convergence and \(\tau ^c_K\)-convergence in \(G\) coincide, (c) \(\tau \)-convergence and \(\tau ^c_H\)-convergence in \(G\) coincide. The authors follow the paper of \textit{P. Piccione} and \textit{R. Sampalmieri} [Commentat. Math. Univ. Carol. 36, No. 3, 551-562 (1995; Zbl 0844.54010)], whose results they complete and improve. The topic of this article is going back to the area of functional differential equations.