Discuss the continuity of the function $f$ defined by $f(x)=\frac{1}{x}, x \neq 0$.

Step 1: Understand the definition of continuity

For a function to be continuous at a point $c$, three conditions must be met: first, $f(c)$ must be defined; second, the limit of $f(x)$ as $x$ approaches $c$ must exist; and third, these two values must be equal. If any of these conditions are not met, the function is discontinuous at that point.

Step 2: Analyze the domain of the function

The given function is $f(x) = \frac{1}{x}$. The problem statement explicitly states that $x \neq 0$. This means the function is not defined at $x=0$. The domain of the function is all real numbers except $0$, i.e., $(-\infty, 0) \cup (0, \infty)$.

Step 3: Check continuity at points in the domain

Let $c$ be any real number such that $c \neq 0$. For such a $c$, $f(c) = \frac{1}{c}$ is defined. Also, the limit of $f(x)$ as $x$ approaches $c$ is $\lim_{x \to c} \frac{1}{x} = \frac{1}{c}$. Since $\lim_{x \to c} f(x) = f(c)$, the function is continuous at every point in its domain.

Step 4: Check continuity at $x = 0$

At $x=0$, the function $f(x) = \frac{1}{x}$ is undefined. According to the definition of continuity, a function cannot be continuous at a point where it is not defined. Therefore, the function is discontinuous at $x=0$.

Step 5: Conclusion on continuity

Based on our analysis, the function $f(x) = \frac{1}{x}$ is continuous at every point in its domain, which includes all real numbers except $0$. It is discontinuous at $x=0$ because it is not defined there.

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