https://file.aiquickdraw.com/mathbot/file/172565687521427ct8y7f.jpg

The image shows a graph of a function \( f(x) \) and asks questions about the continuity of the function at given points, including both visual and numerical parts.

Here’s how to address the questions step by step:

### Part (a): Checking continuity at the given points
For each point, you must decide if \( f(x) \) is continuous:

1. **At \( x = -2 \)**: 
   - By looking at the graph at \( x = -2 \), we need to check whether the function value and the limit from both sides exist and are equal.
   
2. **At \( x = 2 \)**: 
   - Similarly, check the function behavior at \( x = 2 \) to see if it's continuous or has any jump or removable discontinuity.

3. **At \( x = 4 \)**:
   - Check if the function behaves smoothly at \( x = 4 \), ensuring the left-hand and right-hand limits match the function value.

### Part (b): Determining one-sided continuity
This part requires analyzing if the function is continuous from the left, right, or both at each given point:

1. **At \( x = -2 \)**: 
   - Is \( f(x) \) continuous from the left (\( l \)) or the right (\( r \))?

2. **At \( x = 2 \)**:
   - Check whether the function is continuous from the left, right, or both at \( x = 2 \).

3. **At \( x = 4 \)**:
   - For \( x = 4 \), see if the function is continuous from the left, right, or both.

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### Let’s summarize and provide some tips:

Would you like a detailed breakdown of each point for continuity analysis and the corresponding side continuity? I can walk through each part of the function behavior for clarity.

#### Related Questions:
1. What are the conditions for a function to be continuous at a point?
2. What is the difference between one-sided and two-sided continuity?
3. How do you identify discontinuities on a graph?
4. What is the concept of a removable discontinuity?
5. Can you explain jump discontinuities and their implications?

#### Tip:
When determining continuity from a graph, always check the function’s value and limits from both sides at the point of interest.

Determine if f(x) is continuous at specific points (-2, 2, and 4), and from which sides (left or right) the function is continuous.

Analyze the continuity of the function f(x) at x = -2, 2, and 4, using limit laws and the definition of continuity. This problem also explores one-sided continuity, determining if the function is continuous from the left, right, or both at each point. Suitable for students in Grades 10-12.

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