A function f: [a, b] → R is said to satisfy a Lipschitz condition with Lipschitz constant L on [a, b] if, for every x, y ∈ [a, b], we have |f (x) − f (y)| ≤ L|x − y|.
a. Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an interval [a, b], then f ∈ C[a, b].
b. Showthat if f has a derivative that is bounded on [a, b] by L, then f satisfies a Lipschitz condition with Lipschitz constant L on [a, b].
c. Give an example of a function that is continuous on a closed interval but does not satisfy a
Lipschitz condition on the interval