TITLE: What are the points of discontinuity of $\tan x$?
QUESTION [10 upvotes]: $f(x) = \tan x$ 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.

REPLY [3 votes]: There are no “right” or “wrong” definitions, but there are “standard” and “non-standard” definitions. In my opinion, the definition of continuity from your textbook is non-standard when applied to functions defined on sets with isolated points. For example, the function $f:{\mathbb Z} \to {\mathbb R}$ defined by $f(x) = x$ is a continuous function according to the standard definition but it is discontinuous according to the definition from your textbook.
Here is a standard definition.
Let $D \subset {\mathbb R}$ and $f:D \to {\mathbb R}$. We say that $f$ is continuous at point $c$ if one of the following condition holds:

$c$ is a limit point of $D$ and $\lim_{x\to c} f(x) = c$, or
$c$ is an isolated point of $D$.

Some textbooks define essential discontinuities even for points in $\bar D$ as follows.
Let $E \subset {\bar D}$. We say that $c\in E$ is an essential discontinuity of $f$ on $E$ if there is no function $\hat f: E \to {\mathbb R}$ such that

${\hat f}(x) = f(x)$ for $x\in D\cap E$,
$\hat f(x)$ is continuous at point $c$.

According to this definition, $\pi/2$ is an essential discontinuity of $\tan x$ on $\mathbb R$. Of course, it's important what the set $E$ is, in this definition. For example, consider the the function $g(x):{\mathbb R} \setminus {\mathbb Z}\to \mathbb R$ defined by $g(x) = x - \lfloor x \rfloor$. Then 0 is not an essential discontinuity of $g(x)$ on $[0, 1)$, nor on $(-1,0]$. But 0 is an essential discontinuity of $g(x)$ on $(-1,1)$.