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 YX to preserve the continuity of the I -limit function. Ideal versions of weak exhaustiveness, semi-exhaustiveness, semi-uniform convergence, -convergence and semi- convergence of sequences of functions are introduced. Their relationships are clarified. Assume that a sequence of functions \fₙ\₍ ₍ pointwise I -converges to f, we prove that: (a) f is continuous if and only if the sequence \fₙ\₍ ₍ is weakly I -exhaustive. (b) If the sequence \fₙ\₍ ₍ is semi- I -exhaustive, then f is continuous. (c) If the sequence \fₙ\₍ ₍ semi-uniformly I -converges to f and fₙ is continuous for every n N, then f is continuous. (d) If I is? good? and X is first countable, then \fₙ\₍ ₍ is I - convergent to f if and only if \fₙ\₍ ₍ is I -exhaustive. (e) If the sequence \fₙ\₍ ₍ semi- I - converges to f, then f is continuous.
Zhong et al. (Wed,) studied this question.