Question

find the intervals on which the function is continuous.
y = \frac{2}{x^{2}-16}
discontinuous only when x = -4
discontinuous only when x = 16
discontinuous only when x = -4 or x = 4
discontinuous only when x = -16 or x = 16

Explanation:

Step1: Recall continuity condition

A rational function is discontinuous where the denominator is zero.
Set $x^{2}-16 = 0$.

Step2: Solve the equation

We have $x^{2}-16=(x + 4)(x - 4)=0$.
Using the zero - product property, if $(x + 4)(x - 4)=0$, then $x+4 = 0$ or $x - 4=0$.
So $x=-4$ or $x = 4$.

Answer:

C. discontinuous only when $x=-4$ or $x = 4$