How do you prove that the function #x*(x-2)/(x-2)# is not continuous at x=2?

1 Answer
Feb 15, 2016

When #x=2# the expression #x*(x-2)/(x-2)# would require division by #0# (which is not defined).

Explanation:

For a function #f(x)# to be continuous at a point #x=a#,
three conditions must be met:

  1. #f(a)# must be defined.
  2. #lim_(xrarra)f(x)# must exist (it can not be inifinite)
  3. #lim_(xrarra)f(x)=f(a)#

Since for #f(x)=x*(x-2)/(x-2)# does not meet condition 1 when #x=2#, it is not continuous at this point.

Note that it is possible to define a function similar to this which is continuous:
#f(x){(=x*(x-2)/(x-2),,("if "x!=2)),(=x,,("if "x=2)):}#