574 Chapter 9 • Sequences and Series of Functions
Definition 9.4.8 A continuous function f : [a, b] ---+ IR is said to be piece-
wise linear^15 if j partition P = {x 0 ,x 1 ,x 2 ,-· · ,xn} of [a,b] and constants
a1, a2, · · · , an, m1, m2, · · · , m n E IR such that Vt= 1, 2, · · · , n ,
t E [xi-1> Xi]=? f(t) = ai + mi(t - Xi-I)·
Remarks 9.4.9 For a given piecewise linear, continuous f defined in 9.4.8,
i
(a) Vi= 1, 2, · · · , n, f(xi-1) = ai and f(xi) = f(a) + I: mk(xk - Xk-1).
k=l
(b)
f(xi) - f(Xi-l)
V i = 1, 2, · · · , n, mi =.
Xi - Xi-l
(c) For convenience later, we define mo= 0.
(d) Geometrically, the graph off consists of line segments connecting the
endpoints (xo, f(xo)), (xi, f(x1)), · · · , (x n , f(xn)), with slopes m1, m2, · · · , mn
respectively.
Thus, the graph of a piecewise linear continuous function is a polygo-
nal arc, and such functions are often called polygonal functions. It was
Lebesgue's genius to see that continuous functions could be approximated by
polygonal functions, and that they in turn can be approximated by polynomials.
y
Figure 9.13
Theorem 9.4.10 If f is continuous on [a, b], then V c > 0, :J polygonal g on
[a, b] 3 II! -gll < c. (The polygonal functions are dense in C[a, b].)
- In approximation theory, a piecewise linear continuous function is called a "spline of
degree one."
