Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$. Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy: $$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$
Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$.
Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy:
$$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$