Let f be a lipschitzian function on x and e a subset of

Question

Let f be a Lipschitzian function on X and E a subset of X where f L on X is the same as for its restriction to E . Let x ∈ X \E . Does f on E uniquely determine f on X?

(a) Prove that the answer is always yes if x is in the closure E of E.

(b) Give an example where the answer is yes for some f even though x is not in E.

(c) Give an example of X, E, f , and x where the answer is no.