On the $L^1$ and pointwise divergence of continuous functions

For a family of continuous functions $f_1,f_2,\dots \text{\hspace{0.17em}⁣}:I\to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$I_f:=\{x\in I\text{\hspace{0.17em}⁣}:\underset{n\to \mathrm{\infty }}{\lim }{f}_n(x)=+\mathrm{\infty }\}\mathrm{.}$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$\underset{n\to \mathrm{\infty }}{\lim }{\int }_x^y{f}_n(t)dt=+\mathrm{\infty }\text{ for all }x,y\in I\text{ with }x<y\mathrm{.}$$ The origin of this problem is the limit behaviour of quasiarithmetic means.