How do you remove a removable discontinuity?

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Answer 1 · 2015-09-20T23:24:43

Please see the explanation section.

Function #f# has a removable discontinuity at #x=a# if #lim_(xrarra)f(x) = L# (for some real number #L#)

But #f(a) !=L#

We "remove" the discontinuity at #a#, by defining a new function as follows:

#g(x) = { (f(x),if,x != a),(L,if,x=a) :}#

For all #x# other than #a#, we see that #g(x) = f(x)#. and #lim_(xrarra)g(x) = L = g(a)#

So #g# is continuous at #a#.

(In more ordinary language, g is the same as f everywhere except at x = a, and g does not have a discontinuity at a.)