continuity: problem 64 in practice book

[alamps3]

64) Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true?
I. There is a constant C>0 st |f(x) - f(y)| <= C for all x and y in [0,1]
II. There is a constant D>0 st |f(x) - f(y)| <= 1 for all x and y in [0,1] that satisfy |x-y\<=D
III. There is a constant E>0 st |f(x) - f(y)| <= E|x-y| for all x, y in [0,1]

answer is I and II only.

Can someone provide a counterexample to 3 to show why it doesn't have to be true.

[alamps3]

how bout f(x) = sqrt(x)

[RiverMyst]

I is true because f(x) is continuous on a compact set, hence it is totally bounded below and above. II is true because it is the definition of epsilon-delta continuity.

III is FALSE. Here is why. f(x) is continuous on a compact set hence it is uniformly continuous. However, it is well known that uniform continuity does not imply Lipschitz continuity, which is condition III, i.e. f(x) = sqrt(x)