91Ó°ÊÓ

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

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,

$f\left(x\right)=2\tan x$

We know that it is a tangent function whose domain is the set of all real numbers except odd integer multiples of $\frac{\mathrm{Ï€}}{2}$written as:

role="math" localid="1647061391433" $\left\{x∈R|x≠\frac{\mathrm{π}}{2}+n\mathrm{π}.\mathrm{n}∈\mathrm{Z}\right\}$

According to the properties of continuities, if a function is either a tangent or secant 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 asymptotes at odd integer multiples of$\frac{\mathrm{Ï€}}{2}$

This implies that at odd integer multiples of $\frac{\mathrm{Ï€}}{2}$f are undefined.

Therefore, the function is continuous for all real numbers except$x=\frac{\mathrm{Ï€}}{2}+n\mathrm{Ï€}$