Question

Let f
f
be the function given by f(x)=∣∣x2−3∣∣⋅(x+0.5)(x2−3)(x+0.5)
f
(
x
)
=
|
x
2
−
3
|
⋅
(
x
+
0.5
)
(
x
2
−
3
)
(
x
+
0.5
)
. On which of the following open intervals is f
f
continuous?

Answer 1

The function $|x^2-3|$ is continuous on $\mathbb R$, because it is composition of continuous on $\mathbb R$ functions $x^2-3$ and $|x|$ (that is $|x^2-3|=g(h(x))$ , where $h(x)=x^2-3$ and $g(x)=|x|$).

So $|x^2-3|(x+0.5)(x^2-3)(x+0.5)$ is a continuous on $\mathbb R$ function, because it is the product of the 4 continuous on $\mathbb R$ functions.

Since $|x^2-3|(x+0.5)(x^2-3)(x+0.5)$ is a continuous function on $\mathbb R$, it is continuous on all subsets of $\mathbb R$.