How to prove that #f(x):=|x|/x#, if #x!=0# and #f(x)=0#, if #x=0#, is not continuous at 0 ?

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Answer 1 · 2017-06-07T11:56:51

Please see below.

Recall that #absx = {(x,x >= 0),(-x,x < 0):}#

Therefore, #absx/x = {(x/x=1,x > 0),(-x/x=-1,x < 0):}#

Hence,

#lim_(xrarr0^+)f(x) = 1# and #lim_(xrarr0^-)f(x) = -1# .

Because the one-sided limits are not equal, the limit at 0 does not exist.

Therefore #f# is not continuous at #0#.