According to wolfram a function is monotonic if its derivative never changes sign, but the derivative doesn't have to be continuous. So I feel the answer is Yes, tangent is monotonically increasing. Maybe not?
[Math] Is tangent monotonically increasing
functions
Best Answer
Any function defined from $\Bbb R$ (the set of real numbers) to $\Bbb R$ is monotonic iff its derivative never changes sign, yes. But $\tan(x)$ is not a function from $\Bbb R$ to $\Bbb R$, since it's not defined on all real numbers.
In fact, the thing about derivatives is only true when the domain is a connected set (i.e. an interval)! (Remember that $\Bbb R=(-\infty,\infty)$ and is thus an interval.)
The domain of $\tan(x)$ is: $$\left\{x\in\Bbb R:x\ne k\pi+\frac\pi2\right\}$$ This is not an interval! Therefore, the theorem linking derivatives to monotonicity does not hold.
In fact, $\tan(x)$ is not monotonic. To see this, note that: $$\frac\pi3<\frac{2\pi}3$$ but: $$\tan\Big(\frac\pi3\Big)>\tan\Big(\frac{2\pi}3\Big)$$ (The former is $\sqrt3$; the latter is $-\sqrt3$.)
However, $\tan(x)$ is monotonic over the interval $(-\pi/2,\pi/2)$. And it is monotonic over any interval on which it's defined. But it's not monotonic over its entire domain.