How do you determine removable discontinuity for a function?

1 Answer
Sep 25, 2014

Recall that a function #f(x)# is continuous at #a# if

#lim_{x to a}f(x)=f(a)#,

which can be divided into three conditions:

C1: #lim_{x to a }f(x)# exists.
C2: #f(a)# is defined.
C3: C1 = C2

A removable discontinuity occurs when C1 is satisfied, but at least one of C2 or C3 is violated. For example, #f(x)={x^2-1}/{x-1}# has a removable discontinuity at #x=1# since

#lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =lim_{x to 1}(x+1)=2#,

but #f(1)# is undefined.