  We study Orlicz functions that do not satisfy the $\Delta_2$-condition at
zero.
  We prove that for every Orlicz function $M$ such that
$\limsup_{t\to0}M(t)/t^p >0$ for some $p\ge1$, there exists a positive sequence
$T=(t_k)_{k=1}^\infty$ tending to zero and such that
  $$
  \sup_{k\in\mathbb{N}}\frac{M(ct_k)}{M(t_k)} <\infty,\text{ for all }c>1,
  $$
  that is, $M$ satisfies the $\Delta_2$ condition with respect to $T$.
  Consequently, we show that for each Orlicz function with lower Boyd index
$\alpha_M < \infty$ there exists an Orlicz function $N$ such that:
  (a) there exists a positive sequence $T=(t_k)_{k=1}^\infty$ tending to zero
such that $N$ satisfies the $\Delta_2$ condition with respect to $T$, and
  (b) the space $h_N$ is isomorphic to a subspace of $h_M$ generated by one
vector.
  We apply this result to find the maximal possible order of G\^ateaux
differentiability of a continuous bump function on the Orlicz space
$h_M(\Gamma)$ for $\Gamma$ uncountable.
