Using the definition of continuity, prove f (x) = x2 is continuous at x =3.

To prove that the function \( f(x) = x^2 \) is continuous at \( x = 3 \) using the **definition of continuity**, we need to show that for any given \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - 3| < \delta \), it follows that \( |f(x) - f(3)| < \epsilon \).

### Step-by-Step Proof:

1. **Definition of Continuity:**
   A function \( f(x) \) is continuous at \( x = 3 \) if:
   \[
   \lim_{x \to 3} f(x) = f(3)
   \]
   This is equivalent to saying that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - 3| < \delta \), then \( |f(x) - f(3)| < \epsilon \).

2. **Substitute the Function \( f(x) = x^2 \):**
   We need to find \( \delta \) such that:
   \[
   |f(x) - f(3)| = |x^2 - 9| < \epsilon
   \]

3. **Simplify the Expression:**
   We can factorize \( |x^2 - 9| \) as:
   \[
   |x^2 - 9| = |(x - 3)(x + 3)|
   \]
   So, the inequality we need to satisfy becomes:
   \[
   |(x - 3)(x + 3)| < \epsilon
   \]

4. **Establish a Bound on \( |x + 3| \):**
   To find a \( \delta \), note that if \( |x - 3| < \delta \), then \( x \) is close to 3. Specifically, we can bound \( |x + 3| \):
   \[
   |x + 3| \leq |3 + 3| + |x - 3| = 6 + \delta
   \]
   This means \( |(x - 3)(x + 3)| \leq \delta(6 + \delta) \).

5. **Choose \( \delta \):**
   We want \( \delta \) to satisfy:
   \[
   \delta(6 + \delta) < \epsilon
   \]
   To simplify the choice of \( \delta \), we can choose \( \delta \) small enough so that \( 6 + \delta \approx 6 \). This approximation gives:
   \[
   \delta \times 6 < \epsilon \quad \Rightarrow \quad \delta < \frac{\epsilon}{6}
   \]

6. **Conclusion:**
   Thus, for any given \( \epsilon > 0 \), we can choose \( \delta = \min\left(1, \frac{\epsilon}{6}\right) \). This ensures that whenever \( |x - 3| < \delta \), the condition \( |x^2 - 9| < \epsilon \) is satisfied.

Therefore, by the definition of continuity, \( f(x) = x^2 \) is continuous at \( x = 3 \).

### Do you have any questions or need further details?

### 5 Related Questions:
1. How would you prove continuity for other points, such as \( x = 0 \)?
2. What is the geometric interpretation of continuity for polynomial functions?
3. Can you apply this method to non-polynomial functions?
4. How does this proof differ when proving differentiability at a point?
5. What role does the choice of \( \delta \) play in determining the continuity of a function?

### Tip:
When proving continuity, carefully choose \( \delta \) in relation to \( \epsilon \) and make sure it holds for the entire domain around the point of interest.

Learn how to prove the continuity of the function f(x) = x^2 at x = 3 using the definition of continuity. This detailed mathematical proof involves understanding limits, functions, and the definition of continuity theorem. Suitable for advanced high school students and those studying calculus.

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