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.
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.