To show that the function $f(x) = \frac{3x - 1}{x}$ is continuous, we need to verify that it satisfies the definition of continuity.

Definition of Continuity:

A function $f(x)$ is said to be continuous at a point $x = c$ if the following three conditions hold:

1. The function is defined at $x = c$, i.e., $f(c)$ exists.
2. The limit of the function exists as $x$ approaches $c$, i.e., $\lim_{x \to c} f(x)$ exists.
3. The limit is equal to the value of the function at that point, i.e., $\lim_{x \to c} f(x) = f(c)$.

Step-by-step Analysis:

1. Domain of the function: The function is defined as $f(x) = \frac{3x - 1}{x}$. The denominator $x$ cannot be zero, so the function is undefined at $x = 0$. Therefore, the domain of the function is all real numbers except $x = 0$.

2. Continuity at points in the domain: For $x \neq 0$, the function is a rational function (the ratio of two polynomials). Rational functions are continuous wherever they are defined. This means that $f(x)$ is continuous at all points except where the denominator is zero, which only occurs at $x = 0$.

3. At $x = 0$: At $x = 0$, the function is not defined because the denominator is zero, so the function cannot be continuous at this point. However, this doesn't affect the continuity of the function at other points in its domain.

4. Conclusion: Since $f(x)$ is continuous at all points where it is defined (i.e., $x \neq 0$), we conclude that $f(x) = \frac{3x - 1}{x}$ is continuous for all $x \neq 0$.

Tip:

For rational functions like $f(x) = \frac{3x - 1}{x}$, always check for points where the denominator might be zero. These points are where the function might be discontinuous.

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Would you like further details, or do you have any questions?

Here are 5 related questions:

1. How do you determine if a function is discontinuous?
2. What are the types of discontinuities a function can have?
3. Can you explain the difference between pointwise and uniform continuity?
4. How do limits relate to the continuity of a function?
5. How would you prove the continuity of a piecewise function?