88 CHAPTER 3. STONE WEIERSTRASS APPROXIMATION THEOREM
9. Show using the Weierstrass approximation theorem that if f is a continuous, realvalued function on [a,b] , then it has an antiderivative. Hint: Let pn→ f uniformlyand let P′n = pn,Pn (a) = 0. It is obvious that a polynomial has an antiderivative.Now use the uniform convergence of the pn and the mean value theorem from singlevariable calculus to show that {Pn} also converges uniformly to some function F andthat F is the desired antiderivative.
F (x+h)−F (x)h
=F (x+h)−Pn (x+h)
h+
Pn (x+h)−Pn (x)h
+Pn (x)−F (x)
h
= εn(h) (h)+Pn (x+h)−Pn (x)
h= εn(h) (h)+ pn (x+θ hh)
= εn(h) (h)+(pn (x+θ hh)− f (x+θ hh))+( f (x+θ hh)− f (x))
Here n(h) is so large that limh→0 εn(h) (h) = 0. Now pass to a limit.
10. In the above problem, explain, using the mean value theorem from calculus, howyou could define
∫ ba f (x)dx≡ F (b)−F (a) where F is an antiderivative, and thereby
obtain the integral of elementary calculus.