Question

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

$f\left(x\right)=4\cos ecx$

Short answer

f is continuous at all real numbers except $x=n\mathrm{\pi }$.

Step-by-step solution

Step 1. Given information 

We have been given a function $f\left(x\right)=4\cos ecx$.

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,

$f\left(x\right)=4\cos ecx$

We know that it is a cosecant function whose domain is the set of all real numbers except odd integer multiples of $\mathrm{\pi }$, written as :

$\left\{x\in R|x\ne n\mathrm{\pi }.\mathrm{n}\in \mathrm{Z}\right\}$

Step 4. Determine the continuity of the function

According to the properties of continuities, if a function is either a cotangent or cosecant 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 except at the vertical asymptotės at integer multiples of $\mathrm{\pi }$.

This implies that at integer multiples of $\mathrm{\pi }$, f is undefined.

Therefore, the function is continuous for all real numbers except$x=n\mathrm{\pi }$