Testing for Continuity of a piecewise function

mikkicastor05

New member
Joined
Sep 3, 2011
Messages
1
Using the Larson 8th Calculus textbook (Ch1.4, pg. 76), it's discussing the intervals on which the piecewise function

g(x) = sin(1/x), x is not equal to 0
0, x=0

is continuous. It says the intervals of continuity are (-inf, 0) and (0, inf). I understand why sin(1/x) does not have continuity at 0.

My question is, why does the second part of the piecewise (g(x)=0, x=0) not make the entire function continuous on the entire real line?
 

Attachments

  • Untitled1.png
    Untitled1.png
    2.7 KB · Views: 3
It seems to me that you might be thinking something like "If 1/0 were defined, then sin(1/0) would be 0 because sin(0) is 0".

Graph the function sin(1/x) from x = -0.1 to 0.1 and look what is happening near x = 0.

In order for a function to be continuous at x = 0, one of the criteria is that the function must be approaching the same fixed value as x moves toward zero from both directions.

Is it?
 
Top