GR9768 Q 64 Option 3? slope bounded?

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

GR9768 Q 64 Option 3? slope bounded?

Postby Hom » Tue Oct 11, 2011 7:00 am

Here is a old post regarding this questions below
(viewtopic.php?f=1&t=402&p=1799&hilit=gr9768+64#p1799)

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

The answer rejected III. The discussion said that not all the function are uniformly continuous.

But if we rewrite the condition as:
|f(x) -f(y)| / |x-y| = |K| <= E

So the left hand side become the abs of the slope. I am wondering for a continuous function defined on a closed interval, is it possible that the slope in not bounded?

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: GR9768 Q 64 Option 3? slope bounded?

Postby blitzer6266 » Tue Oct 11, 2011 1:11 pm

Any non-Lipschitz function does the job

consider the square root of x...

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: GR9768 Q 64 Option 3? slope bounded?

Postby Hom » Tue Oct 11, 2011 9:10 pm

blitzer6266 wrote:Any non-Lipschitz function does the job

consider the square root of x...


That's a brilliant example. Thank you :D




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