Give an example of a function that is continuous at  x=−2,  x=0 and  x=3 but whose derivative does not exist at any of those points.

An example of a function that is continuous at \( x = -2 \), \( x = 0 \), and \( x = 3 \) but whose derivative does not exist at these points is the absolute value function combined with linear shifts:

\[
f(x) = |x + 2| + |x| + |x - 3|
\]

### Explanation:
1. **Continuity at the points**:
   - The absolute value function \( |x| \) is continuous for all \( x \).
   - Therefore, \( f(x) \) is continuous at \( x = -2 \), \( x = 0 \), and \( x = 3 \).

2. **Derivative at the points**:
   - The derivative of \( |x| \) is not defined at \( x = 0 \) because the slope of the function changes abruptly at this point (it goes from \(-1\) to \(1\)).
   - Similarly, the derivative of \( |x + 2| \) is not defined at \( x = -2 \) and the derivative of \( |x - 3| \) is not defined at \( x = 3 \) for the same reason.

Thus, \( f(x) \) is continuous at \( x = -2 \), \( x = 0 \), and \( x = 3 \), but its derivative does not exist at these points.

Would you like further details on how the derivative behaves at these points or have any other questions?

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Here are 5 related questions:

1. How do you formally prove the continuity of a piecewise function?
2. What are other types of points where a function can be continuous but not differentiable?
3. Can you provide a function that is not continuous at some points but has a derivative at other points?
4. What is the difference between a cusp and a corner in a graph, and how does it affect differentiability?
5. How does the absolute value function behave differently from polynomial functions in terms of continuity and differentiability?

**Tip:** The derivative may fail to exist at points where the function has a sharp corner, a cusp, or a vertical tangent.

Explore an example of a function that is continuous at x = -2, x = 0, and x = 3 but whose derivative does not exist at these points. This problem illustrates concepts of continuity, differentiability, and the behavior of the absolute value function in mathematical analysis.

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