Right Screen 4 for LC C2
GIVEN PROBLEM: Find the points of discontinuity for
 1
x0

f ( x)   x  2
 x5 x  0

SOLUTION:
Step 1: Find the intervals of continuity for the functions within the piecewise defined
function.
1

is continuous on (, 2)  (2, ) . Since we only use x < 0 for this
x2
function, the function is continuous on (, 2)  (2, 0) . Thus, the function is
discontinuous at x = –2.

x  5 is continuous on [5, ) . Since we only use x  0 for this function, the
function is continuous on [0, ) . There are no additional points of discontinuity
for this function.
Step 2: Investigate the limit of the function and the value of the function at the endpoints
of the domain intervals. (i.e. use the definition of continuity)
 Find lim f ( x ) and f (0) .
x 0
1) lim f ( x) 
x 0
1
and lim f ( x)  5 , thus lim f ( x)  DNE .
x 0
x 0
2
2) f (0)  5
Since lim f ( x ) does not exist, the function is discontinuous at x = 0.
x 0
Putting all of this together, we conclude the function f ( x) is discontinuous at x  2
and at x  0 . Thus, the intervals of continuity are (, 2)  (2, 0)  (0, ) .
THIS WOULD BE BETTER IF IT WERE A FLASH MOVIE OR SOME
ANIMATED GIF.