Find the numbers at which f is continuous. At which numbers is f discontinuous?

$f\left(x\right)=-3x^3+7$

We have been given a function $f\left(x\right)=-3x^3+7$.

We have to find the numbers at which f is continuous and at which numbers f is discontinuous.

A function f is discontinuous at the number where:

• f is undefined
• the limit does not exist, i.e., its one-sided limits are not equal
• the limit exists, i.e., its one-sided limits equal, but is not equal to the function value

Analyzing the given function,

$f\left(x\right)=-3x^3+7$

We know that it is a polynomial whose domain is the set of all real numbers, written as

$\left\{x|x\in R\right\}$

According to the properties of continuities, if a function is a polynomial function, then the function's graph must be continuous at every number in the domain, which is at all points of on the real number line. This implies that there is nowhere on its graph that follows the conditions for discontinuity.

Therefore, the function is continuous for all real numbers.