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The problem is asking to check the continuity of the given function \( f(x) \), which is defined as follows:

\[
f(x) = \begin{cases} 
\frac{3x^2 + |x|}{x}, & \text{if } x \neq 0, \\
1, & \text{if } x = 0.
\end{cases}
\]

To check the continuity of this function at \( x = 0 \), we need to ensure that the following conditions hold:
1. The function is defined at \( x = 0 \), which it is, since \( f(0) = 1 \).
2. The limit of \( f(x) \) as \( x \to 0 \) exists.
3. The limit of \( f(x) \) as \( x \to 0 \) must be equal to \( f(0) \).

### Step 1: Simplify \( f(x) \) for \( x \neq 0 \)
For \( x \neq 0 \):
\[
f(x) = \frac{3x^2 + |x|}{x}.
\]
We can break it down into two cases for \( |x| \):

- When \( x > 0 \), \( |x| = x \), so:
  \[
  f(x) = \frac{3x^2 + x}{x} = 3x + 1.
  \]
  
- When \( x < 0 \), \( |x| = -x \), so:
  \[
  f(x) = \frac{3x^2 - x}{x} = 3x - 1.
  \]

Thus, the function \( f(x) \) becomes:
\[
f(x) = \begin{cases} 
3x + 1, & \text{if } x > 0, \\
3x - 1, & \text{if } x < 0.
\end{cases}
\]

### Step 2: Calculate the limit of \( f(x) \) as \( x \to 0 \)

- For \( x \to 0^+ \) (from the right):
  \[
  \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (3x + 1) = 1.
  \]

- For \( x \to 0^- \) (from the left):
  \[
  \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (3x - 1) = -1.
  \]

### Step 3: Check the continuity at \( x = 0 \)

The left-hand limit as \( x \to 0^- \) is \( -1 \), and the right-hand limit as \( x \to 0^+ \) is \( 1 \). Since these two limits are not equal, the limit of \( f(x) \) as \( x \to 0 \) does not exist.

Therefore, \( f(x) \) is **discontinuous** at \( x = 0 \), because the second condition (the existence of the limit) is not satisfied.

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Would you like more details, or do you have any other questions?

Here are 5 related questions:
1. What are the different types of discontinuities a function can have?
2. How do you prove the continuity of a piecewise function at a given point?
3. Can a function be continuous everywhere except at a single point? What is an example?
4. How does the absolute value function affect continuity in piecewise functions?
5. How does the derivative of a piecewise function behave at discontinuous points?

**Tip**: For checking continuity, always verify the existence of the limit from both directions and ensure it matches the function value at the point in question.

Xét tính liên tục của hàm số f(x): f(x) = (3x^2 + |x|)/x, nếu x ≠ 0 và f(x) = 1, nếu x = 0.

Learn how to check the continuity of a piecewise function involving absolute value and quadratic terms. This problem walks through limit calculations and checking if the function is continuous at x = 0. Suitable for high school students (Grades 10-12).

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