What kind of help do you want? What definition of "continuous" are you using? Do you know that "f(x) is continuous at x= a if and only if, given a sequence that converges to a, converges to f(a)"? If so then consider a sequence made entirely of rational numbers and a sequence made entirely of irrational numbers, both converging to a.
As far as "differentiable" is concerned, the derivative at x= 0 is defined as . Again, you can look at all sequences that converge to 0.
For (5) an obvious "appropriate function" is x- 1- ln x. What is the derivative of the that? What does that tell you?
Since (6) says "By considering primitives for ", what is a "primitive" (integral) for that function. You might also want to note that and think about the sum of a geometric series. That will allow you to write as a power series. What is the term-by-term integral of that?