Darboux (intermediate value) property

Prove that a function $f$ defined on an interval $\langle a, b\rangle$ is continuous if and only if

1. the function has the Cauchy property (i.e., the image of every interval $\langle p, q\rangle$ contained in $\langle a, b\rangle$ is an interval), and

2. the set $f^{-1}(\left\{ y\right\} )$ is closed for any $y \in \mathbb {R}$.

Let the function $f$ be differentiable on the segment $\left[a, b\right]$. Prove that if $f^{\prime }(a) f^{\prime }(b)<0$, then there exists a point $c \in (a, b)$ such that $f^{\prime }(c)=0$ (Darboux's theorem).

Let the function $f$ be differentiable on the segment $\left[a, b\right]$. Prove that if the set $\left\{ x \in [a, b] \mid f^{\prime }(x)=y\right\}$ is closed for every $y \in \mathbb {R}$, then $f \in C^{1}(\left[a, b\right])$.