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To evaluate the continuity of the given piecewise function at \(x = -3\), we need to check the following conditions:

### 1. The function value at \(x = -3\):
We evaluate \(f(-3)\) using the relevant branch of the piecewise function:

\[
f(x) = 3x + 1 \quad \text{for} \quad x \leq -3
\]

Thus,
\[
f(-3) = 3(-3) + 1 = -9 + 1 = -8
\]

### 2. The left-hand limit as \(x \to -3^-\):
We calculate the limit of the function as \(x\) approaches \(-3\) from the left (\(x < -3\)):

\[
\lim_{x \to -3^-} f(x) = \lim_{x \to -3^-} (3x + 1) = 3(-3) + 1 = -9 + 1 = -8
\]

### 3. The right-hand limit as \(x \to -3^+\):
Next, we calculate the limit of the function as \(x\) approaches \(-3\) from the right (\(x > -3\)):

\[
f(x) = x^2 + 1 \quad \text{for} \quad x > -3
\]

\[
\lim_{x \to -3^+} f(x) = (-3)^2 + 1 = 9 + 1 = 10
\]

### Conclusion:
The function is continuous at \(x = -3\) if the following three conditions hold:
- The function is defined at \(x = -3\): \(f(-3) = -8\),
- The left-hand limit exists and equals the function value: \(\lim_{x \to -3^-} f(x) = -8\),
- The right-hand limit exists and equals the function value: \(\lim_{x \to -3^+} f(x) = 10\).

Since the left-hand and right-hand limits are not equal, we conclude that:

\[
\text{The function is not continuous at } x = -3.
\]

Let me know if you would like any further clarification or details. Here are some related questions you might find interesting:

1. How do you prove that a function is continuous at a point?
2. What is the significance of one-sided limits in continuity?
3. Can a function be discontinuous but still have a limit at a point?
4. How do we define continuity for a function defined on intervals?
5. How can discontinuity at a point affect the behavior of a function in an interval?

**Tip**: When evaluating the continuity of a piecewise function, always ensure that the limits from both sides match the function's value at the point for continuity.

Evalúe la continuidad de la función f(x) = {3x + 1 para x ≤ -3; x² + 1 para x > -3} en el punto x = -3.

Learn how to evaluate the continuity of a piecewise function f(x) at x = -3. This problem involves calculating left-hand and right-hand limits and using the definition of continuity to determine whether the function is continuous at the given point. Suitable for high school calculus students.

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