  For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$
($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set
  $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ 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 $$ \lim_{n \to \infty}
\int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$
  The origin of this problem is the limit behaviour of quasiarithmetic means.
