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

Feb 15, 2016

When $x = 2$ the expression $x \cdot \frac{x - 2}{x - 2}$ would require division by $0$ (which is not defined).

#### Explanation:

For a function $f \left(x\right)$ to be continuous at a point $x = a$,
three conditions must be met:

1. $f \left(a\right)$ must be defined.
2. ${\lim}_{x \rightarrow a} f \left(x\right)$ must exist (it can not be inifinite)
3. ${\lim}_{x \rightarrow a} f \left(x\right) = f \left(a\right)$

Since for $f \left(x\right) = x \cdot \frac{x - 2}{x - 2}$ 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:
$f \left(x\right) \left\{\begin{matrix}= x \cdot \frac{x - 2}{x - 2} & \null & \left(\text{if "x!=2) \\ =x & \null & ("if } x = 2\right)\end{matrix}\right.$