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Topic: ******** Open challenge for JOHN GABRIEL ********
Replies: 4   Last Post: Jun 20, 2014 8:12 PM

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Posts: 918
Registered: 12/15/13
******** Open challenge for JOHN GABRIEL ********
Posted: Jun 20, 2014 10:36 AM
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You're online now, I see from pinging and reading
<87ce0496-f06c-480e-ba59-644eec3d7bc6@googlegroups.com> from less
than 10 minutes ago.

I knew you would find my last post irresistible.

So, JOHN GABRIEL, here are a few questions for you. They are really easy,
let alone for someone who presumes to write a textbook on calculus. My
came up with them, and he's closer in age to 10 than to 20.

Thank you.

Please, no one else answer. (-:

(1) f(x) = x^2 with domain Q (rationals).
[Note that this means that f is undefined elsewhere]

At what x values does f have a limit (i.e., that the limit is defined)?
Over what interval(s) is f continuous?
Ditto, differentiable?

(2) g(x) = sin(x) for all x in Q
g(x) = 2 sin(x) for all x in R-Q

Is this relationship (with domain R) a function?
If so:
At what x values does g have a limit?
Over what interval(s) is g continuous?

(3) h(x) = g^(-1)(x)
Choose an appropriate domain for h.
What is it?
Is h a function?
If so:
At what x values does h have a limit?
Over what interval(s) is h continuous?

(4) The Weierstrass function with domain R is continuous everywhere but
differentiable nowhere. Restricting its domain to any arbitrarily small
interval, the length of the curve is infinite.
Is it possible to define a function with domain R that is continuous
everywhere, and which when its domain is restricted to any arbitrarily small
interval, its curve length is infinite, but which is differentiable
If yes, provide the function.
If no, indicate why not (a proof would be nicer).

A clock is ticking.

There is a prize that depends on how right and how quick you are.


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