$f(x) = \tan x$ is defined from $\mathbb R - \{\frac{\pi}{2} (2n+1) \mid n \in \mathbb Z\}$ to $\mathbb R$. For every $x$ in its domain, $$f(x) = \frac{\sin x}{\cos x}$$ where $\cos x$ is never 0. Thus, (in short) $\tan x$ is defined for all points in its domain. Now the question remains, is $\tan x$ discontinuous at $x = \pi/2$ (which is outside its domain)?
The question arises because the test for continuity in a textbook mentions that $f(x)$ is continuous at $x = c$ when:
- $f(c)$ exists.
- $\lim_{x \to c} f(x)$ exists.
- $f(c) = \lim_{x \to c} f(x)$.
And my teacher says failure of any of the above results in $x = c$ being a point of discontinuity. Yet, according to me, first test above merely tests the point for its domain and should be the criteria for any point of discontinuity too.