1549901369-Elements_of_Real_Analysis__Denlinger_

9.4 *Two Results of Weierstrass 575

Proof. Suppose f : [a, b] --t JR is continuous and c > 0. Since f is uniformly
continuous (why?) on [a,b], :J 6 > 0 3 'V x1,x2 E [a,b], lx1 -x2I < 6 ~ lf(x1)-
f(x2)I <€.Choose n EN 3 b:;;_a < 6, and let Pn = {xo,x1,x2, · · · ,xn} be the
regular partition of [a, b] into n subintervals of length b:;;_a. Define g : [a, b] --t JR
by

f (xi) - f (xi-1)

\;/ t E [Xi-1,Xij, g(t) = f(Xi-i) + (t - Xi-1).

Xi - Xi-1

Then g is a polygonal function on [a, b] and \;/ i = 1, 2, · · · , n , g(xi ) = f(xi)·
(Prove that g is continuous on [a, b].)
Lett E [a, b]. Then t E [xi-l, xi] for some i. Since g is piecewise linear, g(t)
is between g(xi-l) and g(xi); that is , between f(xi_i) and f(xi)· Thus, by the
intermediate value theorem,

:Jc E [xi-1, xi] 3 f(c) = g(t).

Show that this implies lg(t) - f(t)I < €.

Since this holds 'Vt E [a, b], II! - gll <con [a, b]. •

The next step is to express polygonal functions as sums of simpler functions,
which we shall be able to approximate with polynomial functions.

Theorem 9.4.11 Every polygonal f : [a, b] --t JR is a constant plus a linear
combination of functions of the form

(x-c)+=max{O,x-c}=. - '

{

0 if x < c·

x - c if x ~ c.

y

c x

Figure 9.14

Proof. Suppose f is a polygonal function on [a, b]. Using the notation of
Definition 9.4.8, show that

(a) 'Vt E [xo, x1], f(t) = f(xo) + (m1 - mo)(t - xo)

n

= f(a) + .Z::::(mi - m i-1)(t - Xi-1)+.

i=l

(b) 'Vt E [x 1, x2], f(t) = f(x1) + (m1 - mo)(t - xo) + (m2 - m1)(t - x1)

n

= f(a) + .Z::::(mi - mi-1)(t-Xi-1)+.

i=l