Remark 2

A filter on $\mathscr{C}(X; Y)$ can converge pointwise to a continuous function without converging uniformly to this function.

For example, on the interval $I = [0, 1]$, let $u_n$ be the real-valued function which is equal to 0 for $x = 0$ and $2/n \leq x \leq 1$, equal to 1 for $x = 1/n$, and linear in each of the intervals $[0, 1/n]$ and $[1/n, 2/n]$. The sequence $(u_n)$ converges pointwise to 0, but does not converge uniformly to 0 in $I$ (cf. Exercise 6).