Rolle's Theorem and the Mean Value Theorem on a Closed Interval

Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a<b$, let $[a,b]$ be the closed interval with endpoints $a$ and $b$, which is order-convex by transitivity of $\le$, and let $(a,b)$ be the open interval. Every $c\in(a,b)$ satisfies $a<c<b$ with $a,b\in[a,b]$, and is therefore an interior point of $[a,b]$.

Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$, as a map from the subset $[a,b]$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$, and differentiable at every point of $(a,b)$, the value $f'(c)$ being well defined by Uniqueness of the Derivative at an Interior Point.

Then the following hold.

1. (Rolle) If $f(a)=f(b)$, then there exists $c\in(a,b)$ with

$$f'(c)=0 .$$

2. (Mean value) There exists $c\in(a,b)$ with

$$f'(c)\,(b-a)=f(b)-f(a).$$