On I-convergence of sequences of functions

Let X be a topological space, Y a uniform space and I an admissible ideal on the set N of natural numbers. In this paper, we mainly study the conditions that be added to pointwise I -convergence of a sequence of (continuous) functions in Y^X to preserve the continuity of the I -limit function. Ideal versions of weak exhaustiveness, semi-exhaustiveness, semi-uniform convergence, α -convergence and semi- α convergence of sequences of functions are introduced. Their relationships are clarified. Assume that a sequence of functions \{f_n\}n ∈ N pointwise I -converges to f , we prove that: (a) f is continuous if and only if the sequence \{f_n\}n ∈ N is weakly I -exhaustive. (b) If the sequence \{f_n\}n ∈ N is semi- I -exhaustive, then f is continuous. (c) If the sequence \{f_n\}n ∈ N semi-uniformly I -converges to f and f_n is continuous for every n ∈ N , then f is continuous. (d) If I is ?good? and X is first countable, then \{f_n\}n ∈ N is I - α convergent to f if and only if \{f_n\}n ∈ N is I -exhaustive. (e) If the sequence \{f_n\}n ∈ N semi- I - α converges to f , then f is continuous.