Question

Does the function `f(x) = x/(|x|-3)` have any discontinuities?

Answer 1

There are discontinuities when `x=+-3`

Explanation:

We have:

`f(x) = x/(|x|-3)`

Her is a graph of the function. At first appearance, it would look as through `f(x)` has discontinuities when `x=+-3`
graph{x/(|x|-3) [-10, 10, -10, 10]}

Both the numerator and denominator are continuous functions in their own right. The combination leading to the definition of `f(x)` can therefore only have discontinuity when the denominator is zero.

ie there could only be a discontinuity if

`|x|-3 = 0 => |x| = 3 => x=+-3`

Hence, There are discontinuities when `x=+-3`