Shouldn't tan(x) be a continuous function

Question

If a 'function ' is continuous it must have its limit at $a$ equal to $f(a)$. Considering tan(x) one may say that it is continuous for its domain but not a continuous function for all real numbers $\mathbb{R}$.

But isn't saying that wrong? Simply because if we take $\mathbb{R}$ as the input of our function our function is no longer a function because a function is defined when 'each' input value has a well defined output which is not the case for all inputs in $\mathbb{R}$. And thus our function is no more a function and we cannot say if it's a non-continuous 'function'

Answer 1

The domain of the tangent function is $\mathbb{R}\setminus\left(\frac\pi2+\pi\mathbb{Z}\right)$ and it is continuous. Since, if $k\in\mathbb Z$, the limit $\lim_{x\to\frac\pi2+k\pi}\tan x$ doesn't exist (in $\mathbb R$), you cannot extend it to a continuous function from $\mathbb R$ to $\mathbb R$.

Answer 2

You don't take the domain as "input". A function consists of the following information

• A set $X$ on which it is defined
• A set $Y$ where it maps to
• The information how a point $x\in X$ is mapped.

So for example you can define $\tan: (\pi/2,3\pi/2)\to \mathbb R, x\mapsto \tan(x)$ or, if you want to take the maximal domain, $$\tan: \bigcup\limits_{n\in\mathbb Z}(\pi(n+1/2),\pi(n+3/2))\to \mathbb R,\ x\mapsto \tan(x)$$ but $\tan$ just isn't defined in, for example $\pi/2$. So your saying that we can take $\tan$ with "input" all of the reals, is not correct.