How can you remove a discontinuity?

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Answer 1 · 2018-03-11T20:26:30

Please see below.

A discontinuity at #x=c# is said to be removable if

#lim_(xrarrc)f(x)# exists. Let's call it #L#.

But #L != f(c)# (Either because #f(c)# is some number other than #L# or because #f(c)# has not been defined.

We "remove" the discontinuity by defining a new function, say #g(x)#

by #g(x) = {(f(x),"if",x != c),(L,"if",x = c):}#.

We now have #g(x) = f(x)# for all #x != c# and #g# is continuous at #c#,

Example

#f(x) = (x^2-1)/(x-1)# is discontinuous at #x=1#. (#f(1)# does not exist)

But #lim_(xrarr1)f(x) = lim_(xrarr1)(x^2-1)/(x-1)#

# = lim_(xrarr1) (x+1) = 2#

So we remove the discontinuity by defining:

#g(x) = {((x^2-1)/(x-1),"if",x != 1),(2,"if",x = 1):}#.