Composition Operators in Grand Lebesgue Spaces

Let $\Omega$ be an open subset of $\mathbb{R}^n$ of finite measure. Let $f$ be a Borel measurable function from $\mathbb{R}$ to $\mathbb{R}$. We prove necessary and sufficient conditions on $f$ in order that the composite function $T_f[g]=f\circ g$ belongs to the Grand Lebesgue space $L_{p),\theta}(\Omega)$ whenever $g$ belongs to $L_{p),\theta}(\Omega)$. We also study continuity, uniform continuity, H\"older and Lipschitz continuity of the composition operator $T_f [\cdot] $ in $L_{p),\theta}(\Omega)$.