My textbook describes the Lipschitz continuous function as follows:
A Lipschitz continuous function is a function $f$ whose rate of change is bounded by a Lipschitz constant $\mathcal{L}$:
$$\forall \mathbf{x}, \forall \mathbf{y}, |f(\mathbf{x}) - f(\mathbf{y})| \le \mathcal{L} || \mathbf{x} - \mathbf{y} ||_2$$
And the Wikipedia article for Lipschitz continuity defines the Lipschitz continuous function as follows:
Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, where $d_X$ denotes the metric on the set $X$ and $d_Y$ is the metric on set $Y$, a function $f : X \to Y$ is called Lipschitz continuous if there exists a real constant $K \ge 0$ such that, for all $x_1$ and $x_2$ in $X$,
$$d_Y(f(x_1), f(x_2)) \le K d_X(x_1, x_2).$$
Any such $K$ is referred to as a Lipschitz constant for the function $f$. $\dots$
In particular, a real-valued function $f : R \to R$ is called Lipschitz continuous if there exists a positive real constant $K$ such that, for all real $x_1$ and $x_2$,
$${\displaystyle |f(x_{1})-f(x_{2})|\leq K|x_{1}-x_{2}|.}$$
In this case, $Y$ is the set of real numbers $\mathbf{R}$ with the standard metric $d_Y(y_1, y_2) = |y_1 − y_2|$, and $X$ is a subset of $\mathbf{R}$.
So, I just want to confirm: The textbook explanation has just chosen the second metric to be the Euclidean metric, but this does not have to be the case in general, as evidenced by the Wikipedia article? I just want to confirm because the textbook passes this off as a general mathematical statement, which is, in my opinion, misleading.
I would appreciate it if people could please take the time to clarify this.
For functions between Euclidean spaces when no norm/metric is specified it is assumed that you are using that usual distance function. This is a well established convention. Of course it is also possible to discuss Lipschitz continuity w.r.t. other metrics on Euclidean spaces.