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.

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.