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?

2.5 - 3x

The function is continuous on all subsets of real numbers.

Explanation:

The function f (x) = ∣x^2-3∣ * (x+0.5) * (x^2-3) * (x+0.5) is composed by the multiplication of 4 parts. The first one, ∣x^2-3∣, is continuous for all real numbers because is composed by h (g (x)) = |g (x) | and g (x) = x^2-3 where both are continuous for all real numbers. The other 3 parts are the polynomials (x+0.5), (x^2-3) and (x+0.5); all polynomials is continuous for all real numbers. So, f (x), the multiplication of these continuous functions, is in consequence continuous on all subsets of real numbers.