Question

Prove that the function defined by $f\left(x\right)=\tan x$ is continuous function.

Answer 1

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

$f\left(x\right)=\frac{\sin x}{\cos x}$ is defined for all real number except $\cos x=0$

$\Rightarrow x=(2n+1)\frac{\pi }{2}$

Let $p\left(x\right)=\sin x$ and $q\left(x\right)=\cos x$

We know that $\sin x$ and $\cos x$ is continuous for all real numbers

$\Rightarrow p\left(x\right)$ and $q\left(x\right)$ is continuous.

By Algebra of continuous function

If $p\left(x\right)$ and $q\left(x\right)$ both continuous for all real numbers.

then $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is continuous for all real numbers such that $q\left(x\right)\ne 0$

$\Rightarrow f\left(x\right)=\frac{\sin x}{\cos x}$ is continuous for all real numbers such that $\cos x\ne 0$ or $x\ne (2n+1)\frac{\pi }{2}$

Hence, $\tan x$ is continuous at all real numbers except $x=(2n+1)\frac{\pi }{2}$