Is the following function Lipschitz continuous, uniformly continuous, or neither?

Question

1. $f: x \mapsto ax + b$ on $\mathbb{R}$ ($a,b\in\mathbb{R}$)

2. $f: x \mapsto x^2$ on $(0, 1)$

Answer 1

1. $x \mapsto ax + b$ is Lipschitz continuous with constant $a$, and equality holds.

2. $x \mapsto x^2$ is Lipschitz continuous. If $0 < x < y < 1$, then $y^2 - x^2 = (y-x)(y+x)$ can be controlled below what multiple of $y-x$?