Why is the $\tan$ function differentiable even though it is not continuous?

Question

If only continuous functions can be differentiable, then how can the tangent function $\tan$ be differentiated, even though $\tan$ is not a continuous function?

Answer 1

The tangent function is continuous. I suppose that you are confused by the fact that it has singularities at $k\pi+\frac{\pi}{2}$, $k\in\Bbb Z$, but these points are simply not in the domain of the function.

To be more precise: The function

$$ \tan\colon \Bbb R \setminus \{k\pi+\frac{\pi}{2}: k\in\Bbb Z\} \to \Bbb R$$

is continuous and indeed even differentiable.

Answer 2

Actually, $\tan$ is a continuous (and differentiable) function. Keep in mind that its domain is $\mathbb R\setminus\left\{\frac\pi2+k\pi\,\middle|\,k\in\mathbb Z\right\}$. Since it is the quotient of two differentiable functions, it is differentiable.