Question

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

Answer 1

If a function is not defined at some point, the function is not continuous at that point.

Division of `0` by `0` is undefined

The function
`f(x) = x*(x-2)/(x-2)` contains a term
`(x-2)/(x-2)` which is equivalent to `0/0` when `x=2`
and therefore the function is not continuous at `x=2`

Note that, in this case, we have what is called a "removal discontinuity" but it is still a discontinuity.