Continuous in mathimatics

1. Definitions of continuous

Def 1: (continuous) Given a function $f: X\rightarrow \mathbb{R}$, $\forall x_0 \in X$, $\forall \epsilon >0$, $\exists \eta_{x_0} >0$, $\forall x\in X$ such that $|x_0-x|\leq \eta_{x_0}$, we have $|f(x_0) - f(x)|\leq \epsilon$

Def 2: (uniformly continuous) Given a function $f: X\rightarrow \mathbb{R}$, $\forall \epsilon >0$, $\exists \eta >0$, $\forall x_0, x \in X$ such that $|x_0 - x|\leq \eta$, we have $|f(x_0 ) - f(x)|\leq \epsilon$

Def 3: (absolutely continuous) Given a function $f: X\rightarrow \mathbb{R}$, $\forall \epsilon >0$, $\exists \eta >0$, $\forall [x_i,y_i]_{i=1}^N \subseteq X$ such that $\sum_{i=1}^N|y_i - x_i|\leq \eta$, we have $\sum_{i=1}^N|f(y_i) - f(x_i)|\leq \epsilon$

Def 4: (lipschitz continuous) Given a function $f: X\rightarrow \mathbb{R}$, $f$ is said to be $L$ lipschitz continuous if $\forall x_0, x \in X$, we have

$$|f(x_0) - f(x)|\leq L|x_0 - x|$$

From the definitions of different continuous, we have
lipschitz continuous $\subseteq$ absolutely continuous $\subseteq$ uniformly continuous $\subseteq$ continuous

2. Properties of continuous

Theorem 1: Let $f$ be an absolutely continuous function on $[a, b]$, Then $f$ is of bounded variation on $[a, b]$. Consequently, $f'(x)$ exists for almost every $x \in [a, b]$.

From the theorem, we have
absolutely continuous $\subseteq$ bounded variation $\subseteq$ differentiable almost everywhere.
Moreover, we have
Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ absolutely continuous $\subseteq$ bounded variation $\subseteq$ differentiable almost everywhere