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Topic: Re: ******** Open challenge for JOHN GABRIEL ******** REMINDER 0002
Replies: 1   Last Post: Jun 20, 2014 7:17 PM

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Port563

Posts: 680
Registered: 12/15/13
Re: ******** Open challenge for JOHN GABRIEL ******** REMINDER 0002
Posted: Jun 20, 2014 4:10 PM
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You're online now, I see from pinging 197.79.7.220 and reading a post
from you from less than 7 minutes ago.

You've replied to many posts before and after the one below, so it
must have escaped your attention. This is Reminder 0002.

All easy stuff - tick tock, waiting for your solutions. (-:




"Port563" <reader80@eternal-september.org> wrote...

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
student
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?


BONUS QUESTION
(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
somewhere?
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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