Examples and Counterexamples
(c) It is important that your function g be differentiable at 0, as well as on the rest of (-1,1)
(e) and (f) The word "range" means all the values that the function actually hits. It does not mean just "contained in". So the range of sin(x) is [-1,1]. The range of arctan(x) is the open interval (-pi/2. pi/2). (In both cases I am thinking of the domain as all of R).
1a) There are many ways to prove that a function is continuous/differentiable--the definition certainly can work but so can other approaches like AL.
3) The Mean Value Theorem can be a good friend
4) Go back and review the proofs of Theorems 6.2.6 and 6.3.1. In fact, the false proof of Theorem 6.2.6 on page 176 is a good place to review as well. Again, the point is to follow closely how, starting from epsilon, we choose the delta and/or the N and/or the x satisfying |x-c| < delta. Ask yourself, which variable depends on which other variables. Try not to write an x (or a delta or an N) until after you assert clearly in your proof what exactly this variable is and why it exists. (Until you birng it into existence you really shouldn't be talking about it!)
5) The Lagrange Remainder Theorem, as stated in the book is a little hard to use. What you really need is to look at the proof of the Lagrange Theorem to see how to use it for this exercise. The general statement of the theorem in the book says that the c you end up with is in the interval (-1,1), but if you look at the proof you see that the c really comes from the interval (0,1)...
6) There is really nothing to prove in part (b). This is really just an observation, but an important one. Essentially what this does is up the bar for how many discontinuities a function can have and still be integrable.