1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Transform Method 229

We now define what it means for a function to be piecewise continuous and

calculate the Laplace transform of a piecewise continuous function.

DEFINITION A Piecewise Continuous Function

A function f(x) is piecewise continuous on a finite interval [a,b] if

and only if

( i) f ( x) is continuous on [a, b] except at a finite number of points,

(ii) the limits

f(a+) = lim f(x) and f(b-) = lim f(x)

x--+a+ x--+b-

both exist and are finite, and

(iii) if c E (a, b) is a point of discontinuity of f ( x), then the following

limits exist and are finite:

f(c-) = lim f(x) and f(c+) = lim f(x).

X-l'C- X--+c+

When the limits in (iii) are equal, f is said to have a removable discon-

tinuity at c.

When the limits in (iii) are unequal, f is said to have a jump disconti-

nuity at c.

The function graphed in Figure 5.1 is piecewise continuous on [a, b], has a
removable discontinuity at c 1 , and has jump discontinuities at c 2 and c 3.

• 
  
  
  
  • -----·
  
  
  
  

a b

Figure 5.1 Graph of a Piecewise Continuous Function