As a domain expert in mathematical analysis, I'm here to provide you with an in-depth explanation regarding the continuity of the reciprocal function.

Firstly, let's define what a reciprocal function is. The reciprocal function, often denoted as \( f(x) = \frac{1}{x} \), is a function that takes a non-zero value and returns its multiplicative inverse.

Continuity is a fundamental concept in calculus and analysis. A function \( f \) is said to be continuous at a point \( c \) if the following three conditions are met:
1. \( f(c) \) is defined, i.e., \( c \) is in the domain of \( f \).
2. The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
3. The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

Now, let's consider the reciprocal function \( f(x) = \frac{1}{x} \). This function is defined for all real numbers except \( x = 0 \), since division by zero is undefined. Therefore, the domain of \( f \) is \( (-\infty, 0) \cup (0, \infty) \).

To determine the continuity of \( f \), we examine the limit as \( x \) approaches any point \( c \) in the domain of \( f \). For \( c \neq 0 \), the limit \( \lim_{x \to c} \frac{1}{x} \) exists and is equal to \( \frac{1}{c} \), which is \( f(c) \). This is because as \( x \) gets arbitrarily close to \( c \), the function \( f(x) \) gets arbitrarily close to \( f(c) \), satisfying the definition of continuity.

However, at \( x = 0 \), the function \( f(x) \) is not defined, and hence it cannot be continuous there. As \( x \) approaches zero from either the positive or negative side, the value of \( f(x) \) grows without bound (i.e., \( f(x) \) approaches positive or negative infinity), which means the limit does not exist at \( x = 0 \). Therefore, the reciprocal function is discontinuous at \( x = 0 \).

It's also worth noting that the reciprocal function is not only discontinuous at \( x = 0 \) but also differentiable there, since the function has a sharp turn or cusp at that point.

In summary, the reciprocal function \( f(x) = \frac{1}{x} \) is continuous at every point in its domain except at \( x = 0 \), where it is discontinuous due to the undefined nature of the function at that point.

Now, let's proceed with the translation into Chinese.

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A function is not continuous at any point not in its domain. Hence your reciprocal function is continuous at every value of other than , where it is discontinuous. A function is continuous on an interval if and only if it is continuous at every point of the interval.read more >>