Discuss the continuity of the function $f$ defined by $f(x) = \frac{1}{x}, x \neq 0$.
(N/A) Let $c$ be any non-zero real number in the domain of $f$.
We calculate the limit of the function as $x$ approaches $c$:
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} \frac{1}{x} = \frac{1}{c}$.
Now,we evaluate the function at $x = c$:
$f(c) = \frac{1}{c}$.
Since $\mathop {\lim }\limits_{x \to c} f(x) = f(c)$ for all $c \neq 0$,the function $f(x) = \frac{1}{x}$ is continuous at every point in its domain $(x \in \mathbb{R} \setminus \{0\})$.
Therefore,$f$ is a continuous function.
We calculate the limit of the function as $x$ approaches $c$:
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} \frac{1}{x} = \frac{1}{c}$.
Now,we evaluate the function at $x = c$:
$f(c) = \frac{1}{c}$.
Since $\mathop {\lim }\limits_{x \to c} f(x) = f(c)$ for all $c \neq 0$,the function $f(x) = \frac{1}{x}$ is continuous at every point in its domain $(x \in \mathbb{R} \setminus \{0\})$.
Therefore,$f$ is a continuous function.
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