On the sectionwise connectedness of a contingent

Let $X$ be a real normed space and let $f\colon\mathbb R\to X$ be a continuous mapping. Let $\mathrm T_f(t_0)$ be the contingent of the graph $G(f)$ at a point $(t_0,f(t_0))$ and let $S^+\subset(0,\infty)\times X$ be the “right” unit hemisphere centered at $(0,0_X)$. We show that
1. If $\dim X<\infty$ and the dilation $D(f,t_0)$ of $f$ at $t_0$ is finite then $\mathrm T_f(t_0)\cap S^+$ is compact and connected. The result holds for $\mathrm T_f(t_0)\cap\overline{S^+}$ even with infinite dilation in the case $f\colon[0,\infty)\to X$.
2. If $\dim X=\infty$, then, given any compact set $F\subset S^+$, there exists a Lipschitz mapping $f\colon\mathbb R\to X$ such that $\mathrm T_f(t_0)\cap S^+=F$.
3. But if a closed set $F\subset S^+$ has cardinality greater than that of the continuum then the relation $\mathrm T_f(t_0)\cap S^+=F$ does not hold for any Lipschitz $f\colon\mathbb R\to X$.