Let X be a completely regular (Tychonoff) space, and let C(X), U (X), and B1(X) denote the sets of all real-valued functions on X that are continuous, have a closed graph, and of the first Baire class, respectively. We prove that U (X) = C(X) if and only if X is a P-space (i.e., every G δ-subset of X is open) if and only if B1(X) = U (X). This extends a list of equivalences obtained earlier by Gillman and Henriksen, Onuchic, and Iséki. The first equivalence can be regarded as an unconditional closed graph theorem; it implies that if X is perfectly normal or first countable (e.g., metrizable), or locally compact, then there exist discontinuous functions on X with a closed graph. This complements earlier results by Doboš and Baggs on discontinuity of closed graph functions. P-espacios y un teorema incondicional de gráfica cerrada Resumen. Sea X un espacio completamente regular (Tychonoff). Por C(X), U (X) y B1(X) se denotan los conjuntos de funciones reales definidas en X que son continuas, que tienen gráfica cerrada y que son de primera clase de Baire, respectivamente. Se prueba que U (X) = C(X) si y sólo si X es un P espacio (es decir que cada subconjunot G δ de X es abierto) o si y sólo si B1(X) = U (X). Estos resultados extienden una relación de equivalencias obtenidas por Gillman y Henriksen, Onuchic e Iséki. La primera equivalencia es un teorema incondicional de gráfica cerrada; implica que si X es perfectamente normal o cumple el primer axioma de numerabilidad (por ejemplo si es metrizable), o es localmente compacto, entonces existen funciones discontinuas en X con gráfica cerrada. Así se complementan resultados obtenidos por Dobos y Bags sobre discontinuidad de funciones con gráfica cerrada.

Let X be a completely regular (Tychonoff) space, and let C(X), U (X), and B1(X) denote the sets of all real-valued functions on X that are continuous, have a closed graph, and of the first Baire class, respectively. We prove that U (X) = C(X) if and only if X is a P-space (i.e., every G δ-subset of X is open) if and only if B1(X) = U (X). This extends a list of equivalences obtained earlier by Gillman and Henriksen, Onuchic, and Iséki. The first equivalence can be regarded as an unconditional closed graph theorem; it implies that if X is perfectly normal or first countable (e.g., metrizable), or locally compact, then there exist discontinuous functions on X with a closed graph. This complements earlier results by Doboš and Baggs on discontinuity of closed graph functions. P-espacios y un teorema incondicional de gráfica cerrada Resumen. Sea X un espacio completamente regular (Tychonoff). Por C(X), U (X) y B1(X) se denotan los conjuntos de funciones reales definidas en X que son continuas, que tienen gráfica cerrada y que son de primera clase de Baire, respectivamente. Se prueba que U (X) = C(X) si y sólo si X es un P espacio (es decir que cada subconjunot G δ de X es abierto) o si y sólo si B1(X) = U (X). Estos resultados extienden una relación de equivalencias obtenidas por Gillman y Henriksen, Onuchic e Iséki. La primera equivalencia es un teorema incondicional de gráfica cerrada; implica que si X es perfectamente normal o cumple el primer axioma de numerabilidad (por ejemplo si es metrizable), o es localmente compacto, entonces existen funciones discontinuas en X con gráfica cerrada. Así se complementan resultados obtenidos por Dobos y Bags sobre discontinuidad de funciones con gráfica cerrada.

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Abstract. Let A be a closed G δ -subset of a normal space X. We prove that every function f0 : A → R with a closed graph can be extended to a function f : X → R with a closed graph, too. This is a consequence of a more general result which gives an affine and constructive method of obtaining such extensions.

In the first part of the paper we study the sets of boundedness and of convergence and divergence to infinity of sequences of real closed-graph functions. Generalization on ideal convergence of such sequences is discussed. Limits and ideal-limits of sequences of functions with closed graphs are considered in the last part of the article.