Are constant functions Lipschitz?

Question

Are constant functions Lipschitz? If they are, then how do we calculate the Lipschitz constant?

Answer 1

Any $L$ with $|f(x)-f(y)|\le L|x-y|$ for all $x,y$ is a Lipschitz constant for $f$. So if $f$ is constant, any $L\ge0$ is a Lipschitz constant.

Answer 2

Yes, for, if $f$ is a constant function then every $C > 0$ is such that $|f(x) - f(y)| = 0 \leq C|x-y|$ for all suitable $x,y$.