Piecewise Functions

Piecewise Functions

The continuity of piecewise functions must be handled on a case-by-case basis. It is necessary to look separately

A graph often helps determine continuity of piecewise functions, but we should still examine the algebraic representation to verify graphical evidence.

Graph of f(x)

Example: The function

{ x2 x < -1
f(x) = x -1 x < 1
-cos(PIx) x 1

appears in the Field Guide section on piecewise functions. It is continuous at all points with the exception of x=-1, where the "pieces" do not fit smoothly together. Note that f(x) is continuous at x=1 because

lim x = lim cos(PIx) = 1 = f(1).
x --> 1- x --> 1+

Equipment Check: Let g(x) be the piecewise function

{ ex x < 0
g(x) = x + 1 0 x 2
ln(x) x > 2

Decide whether g(x) is continuous at the points x=0 and x=2.

g(x) is continuous at x=0
g(x) is discontinuous at x=0
Check Answer
g(x) is continuous at x=2
g(x) is discontinuous at x=2
Check Answer

 Explanation See Explanation.



© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson