Discuss the continuity of the $sine$ function.

(N/A) To check the continuity of $f(x) = \sin x$,we need to verify if $\mathop {\lim }\limits_{x \to c} f(x) = f(c)$ for any real number $c$.
First,we use the fact that $\mathop {\lim }\limits_{x \to 0} \sin x = 0$.
Let $c$ be any arbitrary real number. We substitute $x = c + h$. As $x \to c$,it follows that $h \to 0$.
Now,we evaluate the limit:
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} \sin x$
$= \mathop {\lim }\limits_{h \to 0} \sin(c + h)$
$= \mathop {\lim }\limits_{h \to 0} [\sin c \cos h + \cos c \sin h]$
$= \sin c \cdot (\mathop {\lim }\limits_{h \to 0} \cos h) + \cos c \cdot (\mathop {\lim }\limits_{h \to 0} \sin h)$
$= \sin c \cdot (1) + \cos c \cdot (0)$
$= \sin c + 0 = \sin c$
Since $\mathop {\lim }\limits_{x \to c} f(x) = \sin c = f(c)$,the function $f(x) = \sin x$ is continuous for all real numbers $c$.