problem with function

[ddn]

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 such that |f(x)-f(y)|<=C for all x and y in [0, 1]
II. There is a constant D>0 such that |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 such that |f(x)-f(y)|<=E|x-y| for all x and y in [0, 1]

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III

I chose (E) but the answer is (C). My argument is that |f(x)-f(y)|/|x-y|<=E. Where is my mistake?

[ddn]

Help me, por favor, El Señor.

[sagiari]

the answer is c because III requires to be differential at all points..which may not be necessary for a continous function...

[ddn]

Thanks sagiari!

[tuanle]

The Answer is C since (III) is wrong. f is a continuous on the closed interval [0,1] then f is bounded (I) and uniformly continuous, but f is not Liptchitz (III).
Eg: f = sqrt(x)

[lmtuan]

Originally Posted by sagiari

the answer is c because III requires to be differential at all points..which may not be necessary for a continous function...

With C, It maybe not differentiable.