Recall that:
a function, #f#, is continuous at a number, #a# if and only if #lim_(xrarra)f(x) = f(a)#.
Furthermore,
#lim_(xrarra)f(x) = f(a)#.if and only if
(i) #lim_(xrarra)f(x)# exists,
(ii) #f(a)# exists , and
(iii) the numbers in (i) and (ii) are equal.
#f# has a removable discontinuity at #a# if and only if #lim_(xrarra)f(x)# exists, but #f# is not continuous at #a#.
This mean that #lim_(xrarra)f(x)# exists, but that #f(a)# either does not exist or #f(a)# is different from the limit.
Discontinuities in general
Many presentations of calculus do not give a precise definition of "#f# has a discontinuity at #a#"
Mathematicians generally mean something like: #f# is defined for some values near #a# (in an open interval containing #a#) though possibly not at #a# and #f# is not continuous at #a#.
(At other times, mathematicians seems to mean #f# is continuous near #a# though possibly not at #a# and #f# is not continuous at #a#.)
For example, The square root function (as a function #RR rarrRR# ) is not continuous at #-6#, but it's not even defined near #-6#. So many mathematicians would not say "the square root function has a discontinuity at #-6#".
