Def 17.3

$f : (x, d) ⟶ (y, d^{'})$ is called Lipschitz
if $d (f (x), f (y)) \leq k d (x, y)$
where $k > 0$ is called Lipschtz constant

Thm pg.143

if $f : A ⟶ R$ is Lipschitz function, $f$ is uniformly continue in $A$

The best(smallest) Lipschitz constant equal to the first order derivative of a differentiable function.

for $f$ continue, $f^{'}$ bounded and $f^{'}$ integrable $⟹$ $f$ Lipschitz

by FT:

f (b) - f (a) = \int_{a}^{b} f^{'} (t) d t \leq \overline{\sum} Δ sup f (ξ) \leq (b - a) k sup f (ξ_{k})

$\exists f^{'}$ and $f^{'}$ is bounded $⟹$ $f$ is Lipschitz

(\exists ξ \in (a, b)) \frac{f ( b ) - f ( a )}{b - a} = f^{'} (ξ)

⟹ f (b) - f (a) = f^{'} (ξ) (b - a) \leq (b - a) sup f^{'}