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Let's break down the problem and explain why the function \( f(x) \) is not continuous at \( x = 0 \).

The function given is:

\[
f(x) = 
\begin{cases} 
x^2 & \text{if } x \neq 0 \\
1 & \text{if } x = 0 
\end{cases}
\]

### Continuity at \( x = 0 \):

For a function to be continuous at a point \( x = c \), it must satisfy three conditions:

1. **\( f(c) \) is defined**.
2. **\( \lim_{x \to c} f(x) \) exists**.
3. **\( \lim_{x \to c} f(x) = f(c) \)**.

Let’s apply these conditions to \( f(x) \) at \( x = 0 \):

- **Condition 1**: \( f(0) \) is defined. 
  From the definition of \( f(x) \), we know \( f(0) = 1 \). So, this condition is satisfied.

- **Condition 2**: \( \lim_{x \to 0} f(x) \) exists.
  Since for \( x \neq 0 \), \( f(x) = x^2 \), we need to check the limit of \( f(x) \) as \( x \to 0 \).
  \[
  \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 = 0
  \]
  The limit exists and equals 0.

- **Condition 3**: \( \lim_{x \to 0} f(x) = f(0) \).
  Here, \( \lim_{x \to 0} f(x) = 0 \), but \( f(0) = 1 \). Since 0 is not equal to 1, this condition is **not satisfied**.

Thus, \( f(x) \) is not continuous at \( x = 0 \) because the limit of \( f(x) \) as \( x \to 0 \) does not equal \( f(0) \).

### Correct Answer:

From the given choices, the correct explanation is:

- **a.** \( \lim_{x \to 0} f(x) \neq f(0) \)

Would you like more details on any step or further clarification?

Here are some related questions to consider:

1. What is the definition of a discontinuous function?
2. How do you calculate the left-hand and right-hand limits?
3. Can a function be discontinuous at multiple points?
4. What happens to a function’s limit if it oscillates near a point?
5. How does the type of discontinuity affect the graph of a function?

**Tip:** When analyzing continuity, always check both the limit and the function value at the given point!

Explain why f(x) = { x^2 if x ≠ 0, 1 if x = 0 } is not continuous at x = 0. Options: a) lim f(x) ≠ f(0), b) lim f(x) = f(0), c) lim f(x) ≠ f(1), d) lim f(x) does not exist.

Understand why the piecewise function f(x) = { x^2 if x ≠ 0, 1 if x = 0 } is not continuous at x = 0 by analyzing its limits and function values. This problem covers key concepts in limits and continuity and is suitable for students in Grades 10-12.

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