GR9768 Q-64

Forum for the GRE subject test in mathematics.
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Joined: Tue Nov 01, 2011 5:44 am

GR9768 Q-64

Postby lifeisgood88 » Tue Nov 01, 2011 4:11 pm

can anyone tell me the solution with full explanation

Suppose f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true?

1. There is a constant C>0 such that |f(x)-f(y)|<= C for all x and y in [0,1]
2. 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
3. There is a constant E>0 such that |f(x)-f(y)|<= E|x-y| for all x and y in [0,1]

plz reply

Posts: 27
Joined: Sun Oct 17, 2010 4:57 am

Re: GR9768 Q-64

Postby fireandgladstone » Tue Nov 01, 2011 9:43 pm

A continuous function on a compact set is bounded. If M is the bound, then C = 2M works for (1). For (2), you have to remember that a continuous function on a compact set is uniformly continuous. (3) isn't true: consider f(x) = x^(1/2).

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