Definition. A function $f$ defined on a set $S \subseteq \mathbb R$ is said to be Lipschitz continuous on $S$ if there exists an $L \geq 0$ such that $$\|f(x_1) - f(x_2)\| \le L\|x_1 - x_2\|$$ for all $x_1$ and $x_2$ in $S$ such that $x_1 \ne x_2$.
Consider some function $f: \mathbb R^n \to \mathbb R^n$ taking several variables $x^{(k)}$ with $1 < k \leq n$ as argument. Let $f_i: \mathbb R^n \to \mathbb R$ be the function of $i$-th coordinate of the function value of $f$. Are the following statements true?
- If $f$ is Lipschitz continuous, then $f_i$ is Lipschitz continuous $\forall i \in \{1, \dots, n\}$.
- If $f_i$ is Lipschitz continuous $\forall i \in \{1, \dots, n\}$, then $f$ is Lipschitz continuous.
$f$ is K-Lipschitz continuous implies $f_i$ is Lipschitz continuous $\forall i \in \{1, \dots, n\}$:
$\left | f_i(x)-f_i(y) \right |\leq \left \| f(x)-f(y) \right \|\leq K\left \| x-y \right \|$
Each coordinate $f_i$ is $K_i$-Lipschitz continuous implies $f$ is Lipschitz-continuous:
$f=\sum_{i=1}^{n}f_i\mathbf{e_i}$ where $\mathbf{e_i}=\left (0,...,0,1,0,...,0 \right )$ where the $1$ is at the $i$-th position.
Note that $\left \| f_1(x)\mathbf{e_1}+f_2(x)\mathbf{e_2}-f_1(y)\mathbf{e_1}-f_2(y)\mathbf{e_2} \right \|\leq \left \| f_1(x)-f_1(y) \right \|+\left \|f_2(x)-f_2(y) \right \|\leq L_1\left \| x-y \right \|+L_2\left \| x-y \right \|\leq (L_1+L_2)\left \| x-y \right \|$
The result follows by induction.