https://file.aiquickdraw.com/mathbot/file/1726710673381qxqnmexc.jpg

The problem asks to express in interval notation where the function \( f \) is differentiable, based on the provided graph.

Differentiability requires that:
1. The function is continuous.
2. The function has no sharp corners (cusps) or vertical tangents.

From analyzing the graph:

### Observations:
1. **Discontinuities**:
   - There is a jump discontinuity at \( x = 1 \). The function is not continuous there, so it cannot be differentiable.
   - There is a removable discontinuity at \( x \approx 3 \) (the open circle). Thus, the function is not differentiable at \( x = 3 \).

2. **Corners**:
   - Sharp corners are visible at \( x \approx 2 \) and \( x \approx 4 \). These points are where the slope of the function changes abruptly, meaning the function is not differentiable at these points.

3. **Vertical Tangent**:
   - There is a vertical tangent at \( x = 0 \), where the slope becomes infinite. The function is not differentiable at \( x = 0 \).

### Differentiable Intervals:
Based on the graph and the observations, the function \( f \) is differentiable on the following intervals:
- From \( (-\infty, 0) \), excluding the discontinuity at \( x = 0 \),
- From \( (0, 1) \), excluding the point at \( x = 1 \),
- From \( (1, 2) \), excluding the sharp corner at \( x = 2 \),
- From \( (2, 3) \), excluding the discontinuity at \( x = 3 \),
- From \( (3, 4) \), excluding the sharp corner at \( x = 4 \),
- From \( (4, 5) \).

### Final Answer in Interval Notation:
The function is differentiable on the intervals:
\[
(-\infty, 0) \cup (0, 1) \cup (1, 2) \cup (2, 3) \cup (3, 4) \cup (4, 5)
\]

Let me know if you'd like more details or have any questions.

### 5 Related Questions:
1. How is differentiability related to continuity?
2. What happens at a sharp corner in terms of differentiability?
3. What is the difference between a removable discontinuity and a jump discontinuity?
4. How do we handle vertical tangents when considering differentiability?
5. How do we find differentiability in piecewise-defined functions?

### Tip:
Always check for both continuity and smoothness (no sharp corners or vertical tangents) when determining differentiability of a function.

Let f be the function in the graph. Express in interval notation where f is differentiable.

Find the intervals where the function f is differentiable based on the given graph, identifying key discontinuities, sharp corners, and vertical tangents. This is suitable for advanced high school or early university-level calculus students.

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