question related to Lipschitz continuous function

Question

Here I am writing what i understood the Lipschitz continuity of a function. Given a function $f:X\rightarrow Y$ defined from a metric space $(X, d_X) $ to a metric space $(Y, d_Y)$ is said to be a Lipschitz continuous function, if there exists at least one real constant $k$ such that $d_Y(f(x_1), f(x_2))\leq k d_X(x_1, x_2),$ $\forall x_1, x_2\in X.$ My questions are (1) Can $k$ may be zero? (2) If there exists more than one $k$ satisfying the above inequality, then which one is a Lipschitz constant? (3) Relation between the Lipschitz continuity of a function, uniform continuity of a function, absolute continuity of a function, differentiability of a function and a function of a bounded variation, boundedness of a function. Here my function $f$ should be defined on any metric space $X$ to $Y$

Answer 1

The other questions have been addressed by comments, so I will only address part (3). Indeed, these relations are addressed by the Wikipedia page for Lipschitz continuous functions. I only want to draw your attention to the fact to the following result:

If $f$ is Lipschitz continuous then it is differentiable almost everywhere.

This general result is known as Rademacher's Theorem.