Question

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

Answer 1

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)):}`