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### Translation of the Question
It is known that the function \( f \) is defined on the entire real line, is continuous at the point \( 0 \), and \( f(0) = 0 \).

What can be said about the continuity of the function
\[
g(x) = 
\begin{cases} 
f(x) + 1, & x \leq 0, \\
f(x) - 1, & x > 0,
\end{cases}
\]
at the point \( x = 0 \)?

Choose one:
1. The function \( g \) is continuous at \( x = 0 \).
2. The function \( g \) is not continuous at \( x = 0 \).
3. There is not enough information to conclude the continuity of \( g \) at \( x = 0 \).

### Solution
To determine if \( g(x) \) is continuous at \( x = 0 \), we need to examine the left-hand limit, the right-hand limit, and the value of \( g(0) \).

1. **Left-hand limit as \( x \to 0^- \):**
   \[
   \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} (f(x) + 1) = f(0) + 1 = 0 + 1 = 1
   \]

2. **Right-hand limit as \( x \to 0^+ \):**
   \[
   \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (f(x) - 1) = f(0) - 1 = 0 - 1 = -1
   \]

3. **Value of \( g(0) \):**
   Since \( g(x) = f(x) + 1 \) for \( x \leq 0 \), we have \( g(0) = f(0) + 1 = 0 + 1 = 1 \).

The left-hand limit and right-hand limit at \( x = 0 \) are not equal (left-hand limit is 1, and the right-hand limit is -1), so \( g(x) \) is not continuous at \( x = 0 \).

### Conclusion
The correct answer is:
- **The function \( g \) is not continuous at \( x = 0 \).**

Would you like more details or have any questions?

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Here are 5 related questions that build on this concept:

1. How would the continuity of \( g(x) \) change if \( f(x) \) were not continuous at \( x = 0 \)?
2. What is the significance of left-hand and right-hand limits in determining continuity?
3. If \( g(x) \) were defined differently for \( x > 0 \), could it still be continuous at \( x = 0 \)?
4. How does the continuity of \( f(x) \) at \( x = 0 \) affect the overall behavior of \( g(x) \)?
5. What are some real-life applications of piecewise functions like \( g(x) \)?

**Tip:** When checking continuity at a point, always compare the left-hand limit, right-hand limit, and the function's value at that point.

It is known that the function \( f \) is defined on the entire real line, continuous at the point \( 0 \), and \( f(0) = 0 \). What can be said about the continuity of the function
\[
g(x) = 
\begin{cases} 
f(x) + 1, & x \leq 0, \\
f(x) - 1, & x > 0,
\end{cases}
\]
at the point \( x = 0 \)?

Explore how to determine the continuity of a piecewise function g(x) defined by conditions involving f(x) at x = 0. This solution covers limit evaluation and continuity concepts suitable for high school students in Grades 11-12.

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