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We study Orlicz functions that do not satisfy the $\Delta_2$-condition atzero.  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 zerosuch 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 onevector.  We apply this result to find the maximal possible order of G\^ateauxdifferentiability of a continuous bump function on the Orlicz space$h_M(\Gamma)$ for $\Gamma$ uncountable.

Orlicz functions that do not satisfy the $\Delta_2$-condition and high order Gateaux smoothness in $ h_M(\Gamma) $