142 CHAPTER 7. THE DERIVATIVE

12. Show that for r a rational number and y = xr, it must be the case that if this functionis differentiable, then y′ = rxr−1.

13. Let f be a continuous function defined on [a,b] . Let ε > 0 be given. Show thereexists a polynomial p such that for all x ∈ [a,b] ,

| f (x)− p(x)|< ε.

This follows from the Weierstrass approximation theorem, Theorem 6.10.3. Nowhere is the interesting part. Show there exists a function g which is also continuouson [a,b] and for all x ∈ [a,b] ,

| f (x)−g(x)|< ε

but g has no derivative at any point. Thus there are enough nowhere differentiablefunctions that any continuous function is uniformly close to one. Explain why everycontinuous function is the uniform limit of nowhere differentiable functions. Alsoexplain why every nowhere differentiable continuous function is the uniform limit ofpolynomials. Hint: You should look at the construction of the nowhere differentiablefunction which is everywhere continuous and bounded, given above.

14. Consider the following nested sequence of compact sets, {Pn}. Let P1 = [0,1], P2 =[0, 1

3

]∪[ 2

3 ,1], etc. To go from Pn to Pn+1, delete the open interval which is the

middle third of each closed interval in Pn. Let P = ∩∞n=1Pn. By Problem 16 on Page

72, P ̸= /0. If you have not worked this exercise, now is the time to do it. Show thetotal length of intervals removed from [0,1] is equal to 1. If you feel ambitious alsoshow there is a one to one onto mapping of [0,1] to P. The set P is called the Cantorset. Thus P has the same number of points in it as [0,1] in the sense that there isa one to one and onto mapping from one to the other even though the length of theintervals removed equals 1. Hint: There are various ways of doing this last part butthe most enlightenment is obtained by exploiting the construction of the Cantor setrather than some silly representation in terms of sums of powers of two and three.All you need to do is use the theorems in the chapter on set theory related to theSchroder Bernstein theorem and show there is an onto map from the Cantor set to[0,1]. If you do this right it will provide a construction which is very useful to provesome even more surprising theorems which you may encounter later if you studycompact metric spaces. The Cantor set is just a simple version of what is seen insome vegetables. Note in the following picture of Romanesco broccoli, the spirals ofpoints each of which is a spiral of points each of which is a spiral of points...

15. ↑ Consider the sequence of functions defined in the following way. Let f1 (x) = x on[0,1]. To get from fn to fn+1, let fn+1 = fn on all intervals where fn is constant. Iffn is nonconstant on [a,b], let fn+1(a) = fn(a), fn+1(b) = fn(b), fn+1 is piecewise