Question

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

$f\left(x\right)=-2\cos x$

Short answer

f is continuous at all real numbers.

Step-by-step solution

Step 1. Given information 

We have been given a function $f\left(x\right)=-2\cos x$.

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

Step 2. Discontinuous function property

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

Step 3. Analyze the function

Analyzing the given function,

$f\left(x\right)=-2\cos x$

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

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

Step 4. Determine the continuity of function

According to the properties of continuities, if a function is a sine or cosine 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.