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

(N/A) The function $f(x) = \tan x = \frac{\sin x}{\cos x}$ is defined for all real numbers $x$ such that $\cos x \neq 0$,which means $x \neq (2n + 1) \frac{\pi}{2}$ for any integer $n$.
We know that both the sine function $g(x) = \sin x$ and the cosine function $h(x) = \cos x$ are continuous for all real numbers.
According to the algebra of continuous functions,if $g(x)$ and $h(x)$ are continuous functions,then their quotient $\frac{g(x)}{h(x)}$ is also continuous at all points where the denominator $h(x) \neq 0$.
Since $f(x) = \frac{\sin x}{\cos x}$ is the quotient of two continuous functions and is defined for all $x \in \mathbb{R} \setminus \{(2n + 1) \frac{\pi}{2} : n \in \mathbb{Z}\}$,it follows that $f(x) = \tan x$ is a continuous function on its entire domain.