572 Chapter 9 • Sequences and Series of Functions
Apply the triangle inequality to get
1
lf(an) - f(x)I + IJ(x) - f(bn)I ;:::: lf(an) - f(bn)I > 2 n ·1 1
Thus, either lf(an) - f(x)I >
2
n+l or lf(x) - f(bn)I >
2
n+l. (Why?) Choose
1
Xn =an or bn, so that lf(xn) - f(x)I >
2
n+l ·
Part 2: Consider the sequence {xn} constructed in Part 1. Show that
lxn - xi ::; -
1- , and so Xn --+ x. But, show that { f(xn) - f(x)} diverges.
lOn Xn - X
Therefore, f is not differentiable at x. (Why?) •
THE WEIERSTRASS APPROXIMATION THEOREMBy the time of Weierstrass (1815- 1897), mathematicians had been making
routine use of infinite series (especially power series) since the creation of cal-
culus some two centuries earlier. They had achieved spectacular successes using
series to represent functions. Mathematicians of Weierstrass' time depended so
heavily on the use of infinite series in their work that they had come to believe
that all functions of any worth could be analyzed using power series.
For a function to have a power series representation it must, of course, have
derivatives of all orders. Even the existence of the Taylor polynomial Tn(x) re-
quires a function to have derivative of order n somewhere. Thus, it became
common for mathematicians to believe that functions, in general, must be dif-
ferentiable at least somewhere. So it is easy to see why Weierstrass' example of
an everywhere continuous, nowhere differentiable function was so disturbing,
even to Weierstrass himself.
In his Approximation Theorem,^11 Weierstrass used a completely orig-
inal approach to show that, given an arbitrary continuous function f on a
compact interval [a, b] and an arbitrary c > 0, there exists a polynomial p such
that
Vx E [a, b], lf(x) - p(x)I < c.
That is, relative to [a, b], II! - Pll < c. To paraphrase, Weierstrass' theorem
guarantees that for any continuous f on [a, b] and any c > 0, there is a poly-
nomial that stays "c-close" to f on [a, b]. For any prescribed accuracy, there is
a polynomial whose values can be used to approximate the values of f(x) with
that accuracy over a given compact interval.
This was a partial vindication of those who advocated the use of (power)
series to represent functions. For in calculations they frequently truncated series
- Published in 1885 , when Weierstrass was 70 years old, thirteen years after he announced
his continuous, nowhe re differentiable function.