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

Apr 7, 2015

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
$f \left(x\right) = x \cdot \frac{x - 2}{x - 2}$ contains a term
$\frac{x - 2}{x - 2}$ which is equivalent to $\frac{0}{0}$ 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.