Darboux property versus continuity

A function $f:(0,1)\to\mathbb{R}$ is said to have property $\mathcal{I}$ if, for any $0<x_1<x_2<1$ and for any $c$ between $f(x_1)$ and $f(x_2)$, there exists $y\in[x_1,x_2]$ such that $f(y)=c$. Consider the following statements: (I) If $g:(0,1)\to\mathbb{R}$ satisfies property $\mathcal{I}$, then $g$ is necessarily continuous. (II) If $h:(0,1)\to\mathbb{R}$ is differentiable, then $h'$ necessarily satisfies property $\mathcal{I}$. Then