3.5. EXERCISES 87

which is uniformly close to f on [0,1]. Extend to the case of a box ∏pk=1 [ak,bk]. The

continuous function f is only required to have values in some normed linear space.

4. Consider the following pm (x)≡

m1

∑k1=0· · ·

mn

∑kn=0

(m1

k1

)(m2

k2

)· · ·(

mn

kn

)xk1

1 (1− x1)m1−k1 xk2

2 (1− x2)m2−k2

· · ·xknn (1− xn)

mn−kn f(

k1

m1, · · · , kn

mn

). (3.3)

where f : [0,1]n→ X a normed linear space. Show that for all ε , there exists n suchthat ∥pm− f∥ < ε if min(m1, · · · ,mn) > 0. Hint: Consider Lemma 3.1.1 first andyou may see how to do this.

5. Theorem 2.5.43 gave an example of a function which is everywhere continuous andnowhere differentiable. The first examples of this sort were given by Weierstrass in1872 who gave an example involving an infinite series in which each term had allderivatives everywere and yet the uniform limit had no derivative anywhere. Usingthe example of Theorem 2.5.43, give an example of an infinite series of functions,each term being a polynomial defined on [0,1], ∑

∞k=1 pk (x)= f (x) for which it makes

absolutely no sense to write f ′ (x) = ∑∞k=1 p′k (x) because f ′ fails to exist at any point.

In other words, you cannot differentiate an infinite series term by term. The deriva-tive of a sum is not the sum of the derivatives when dealing with an infinite “sum”.Also show that if you have any differentiable function g and ε > 0, there exists anowhere differentiable function h such that ∥g−h∥ < ε . This is in stark contrastwith what will be presented in complex analysis in which, thanks to the Cauchyintegral formula, uniform convergence of differentiable functions does lead to a dif-ferentiable function. Hint: Use Weierstrass approximation theorem and telescopingseries to get the example of a series which can’t be differentiated term by term.

6. If f , f ′ are both continuous, suppose pn→ f uniformly where the pn are the Bernsteinpolynomials. Show that then p′n→ f ′ uniformly also.

7. Use the above problem to show that if f is continuous and defined on ∏pk=1 [0,1]

and if also all the partial derivatives of f are continuous, then if pn → f uniformlywith the pn being the Bernstein polynomials discussed in Problem 3, then the partialderivatives of these pn converge uniformly to the corresponding partial derivatives off . Extend to the case where f is defined on ∏

pk=1 [ak,bk].

8. In contrast to Problem 6, consider the sequence of functions

{ fn (x)}∞

n=1 =

{x

1+nx2

}∞

n=1.

Show it converges uniformly to f (x) ≡ 0. However, f ′n (0) converges to 1, notf ′ (0). Hint: To show the first part, find the value of x which maximizes the func-tion

∣∣∣ x1+nx2

∣∣∣ . You know how to do this. Then plug it in and you will have an estimatesufficient to verify uniform convergence. This shows how special the Bernstein poly-nomials are.