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Let's analyze the given piecewise function \( f(x) \):

\[
f(x) = 
\begin{cases} 
7, & x < 6, \\
0, & x = 6, \\
10 + x, & x > 6.
\end{cases}
\]

We need to find the points of discontinuity and evaluate the one-sided limits at each of these points.

### Step 1: Check continuity at \( x = 6 \).

For \( f(x) \) to be continuous at \( x = 6 \), the following conditions must be satisfied:
1. \( \lim_{x \to 6^-} f(x) = \lim_{x \to 6^+} f(x) \) (the left-hand and right-hand limits must exist and be equal).
2. The value of the function at \( x = 6 \) must match the limits: \( \lim_{x \to 6} f(x) = f(6) \).

#### 1. Left-hand limit as \( x \to 6^- \):
When \( x < 6 \), \( f(x) = 7 \).
\[
\lim_{x \to 6^-} f(x) = 7.
\]

#### 2. Right-hand limit as \( x \to 6^+ \):
When \( x > 6 \), \( f(x) = 10 + x \).
\[
\lim_{x \to 6^+} f(x) = 10 + 6 = 16.
\]

#### 3. Value of the function at \( x = 6 \):
\( f(6) = 0 \).

Since \( \lim_{x \to 6^-} f(x) = 7 \) and \( \lim_{x \to 6^+} f(x) = 16 \), these one-sided limits are not equal. Hence, \( f(x) \) is **discontinuous** at \( x = 6 \). Additionally, \( f(6) = 0 \), which does not match either of the limits.

### Conclusion:
- \( x = 6 \) is a point of discontinuity.
- The one-sided limits are:
  - \( \lim_{x \to 6^-} f(x) = 7 \),
  - \( \lim_{x \to 6^+} f(x) = 16 \).

Would you like a deeper explanation on this or any follow-up questions? Here are some relative questions to expand on the topic:

1. How do you formally define the concept of continuity at a point?
2. Can you explain the difference between jump discontinuity and removable discontinuity?
3. What are other types of discontinuities besides jump discontinuity?
4. How do you handle discontinuities in real-world applications?
5. What are some common techniques to evaluate limits involving piecewise functions?

**Tip**: When dealing with piecewise functions, always check the one-sided limits at the points where the definition of the function changes to determine discontinuities.

Find each point of discontinuity of f(x), and for each point of discontinuity, evaluate the indicated one-sided limits. Given f(x) as: f(x) = 7 for x < 6, f(x) = 0 for x = 6, and f(x) = 10 + x for x > 6.

Explore how to find the points of discontinuity and evaluate one-sided limits for a piecewise function. This problem discusses continuity conditions and provides a detailed solution for a function defined in three intervals.

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