Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!) I'm doing a test about limits and continuity and got these two wrong.

$\mathbf{Q1}$: The function $f(x) = \tan x$:
  $\hspace{1em}\mathtt{a)}$ is continuous
  $\hspace{1em}\mathtt{b)}$ is discontinuous
  $\hspace{1em}\mathtt{c)}$ is increasing
  $\hspace{1em}\mathtt{d)}$ is even  

According to my professor the right one is $\mathtt{a)}$, but I don't see how $\tan x$ can be continuous. In fact, my answer was $\mathtt{b)}$, because it's discontinuous at $x = \frac\pi2 + k\pi$, with $k \in \mathbb{Z}$. Is there a definition of continuity according to which $\tan x$ is continuous?

$\mathbf{Q2}$: Between the following, select the correct sentence
  $\hspace{1em}\mathtt{a)}$ if $\displaystyle\lim_{x \to +\infty} f(x) = +\infty$ then $f$ is increasing
  $\hspace{1em}\mathtt{b)}$ if $\displaystyle\lim_{x \to +\infty} f(x) = +\infty$ then there exist an half line $(r, +\infty)$ on which $f$ can be inverted
  $\hspace{1em}\mathtt{c)}$ the fact that $\displaystyle\lim_{x \to +\infty} f(x) = +\infty$ does not imply that $f$'s domain contains half lines of the form $(r, +\infty)$
  $\hspace{1em}\mathtt{d)}$ if $\displaystyle\lim_{x \to +\infty} f(x) = +\infty$ and $f$ can be inverted, then $\displaystyle\lim_{x \to +\infty}f^{-1}(x) = +\infty$

Here the correct one is $\mathtt{c)}$, while I replied $\mathtt{d)}$. I thought that given the premises in $\mathtt{d)}$, $f$ would be monotone increasing and thus $f^{-1}$ would be as well. Examples: $\exp(x)$ and $\ln x$, $x$ and $x$, $x^2$ (with $x > 0$) and $\sqrt{x}$...
Apart from $\mathtt{a)}$, which is obviously wrong, can you explain me the other answers?
 A: The $\tan$ function is defined on $\Bbb R\setminus \left\{\frac{\pi}2+k\pi,\; k\in\Bbb Z\right\}$ and it's continuous on this set. You can't say that this function is discontinuous on $\frac{\pi}2+k\pi$ since it isn't even defined on these points.
A: Ok, so my initial comments were wrong.  A function can be labeled continuous or discontinuous at a point only if it is first defined at the point.  Think about the definition of continuity:
A function $f$ is continuous at $x \in \mathbb{R}$ if:

$\forall \epsilon > 0$ $\exists \delta > 0$ such that for all $y \in \Bbb R$ with $|x - y|<\delta$ it follows that $|f(x) - f(y)| < \epsilon$.  

This definition uses $f(x)$, so $f$ must be defined here.
Similarly, negating the above definition tells us when a function is discontinuous at a point $x \in \mathbb{R}$.  A function $f(x)$ is discontinuous at a point $x \in \mathbb{R}$ if:

$\exists \epsilon > 0$ such that $\forall \delta > 0$ $\exists y \in \mathbb{R}$ with $|x - y| < \delta$ and $|f(x) - f(y)| \geq \epsilon$.  

This definition also uses $f(x)$, so $f$ should be defined at the point $x$ in order to be discontinuous at it.
So, $\tan{x}$ is continuous everywhere that it is defined.  On, for example, $x = \frac{\pi}{2}$, $\tan{x}$ is not defined, and therefore we cannot talk about continuity or discontinuity at this point.
A: About the second question, the answer (d) is valid if you are dealing with a continuous function. Here, the trick is that the function $f$ can be discontinuous.
I remember a good example someone showed me a little while ago. The trick is to construct a function that splits on the sides of the $y$-axis. With such a function (that has to be a bijection) we can clearly see that the $f^{-1}$ function has no limit for ${x \to +\infty}$
Here is one:
$f:\mathbb{R} \to \mathbb{R_+}$
$$f(x) = \begin{cases}2\lfloor x\rfloor + \{x\}&\quad x\ge0\\
2\lfloor -x \rfloor - 1 + \{x\}&\quad x < 0\end{cases}$$
Where $\lfloor x\rfloor$ is the integer part of $x$ and $\{x\}$ is the fractional part of $x$.
Edit : It also work for the question (b) I think. Since $\lim_{x\to +\infty} f(x) = +\infty$ but you cannot chose a half line $(r,+\infty)$ where $f$ can be inverted (because it will be discontinuous on the half line and you will be missing the part before the half line to have it inverted).
