Let's analyze the graphs of $f(x)$ and $g(x)$ to answer the continuity and differentiability questions.

Continuity of $f(x)$:

• At $x = 1$: The graph shows a jump discontinuity at $x = 1$, where the function jumps from an open circle to a filled circle. Therefore, $f(x)$ is not continuous at $x = 1$.
• At $x = 2$: The graph is continuous at $x = 2$, as there are no breaks or jumps. $f(x)$ is continuous at $x = 2$.
• At $x = 3$: Similarly, there is no break in the graph at $x = 3$, so $f(x)$ is continuous at $x = 3$.
• At $x = 4$: The graph is continuous at $x = 4$, with no jump or discontinuity. $f(x)$ is continuous at $x = 4$.

Differentiability of $f(x)$:

• At $x = 1$: Since $f(x)$ is not continuous at $x = 1$, it cannot be differentiable there. $f(x)$ is not differentiable at $x = 1$.
• At $x = 2$: The graph has a sharp corner at $x = 2$, which means the derivative is not defined. $f(x)$ is not differentiable at $x = 2$.
• At $x = 3$: The graph is smooth at $x = 3$, so it is differentiable. $f(x)$ is differentiable at $x = 3$.
• At $x = 4$: There is a sharp corner at $x = 4$, so $f(x)$ is not differentiable at $x = 4$.

Differentiability of $g(x)$:

• At $x = 1$: The graph is smooth at $x = 1$, so $g(x)$ is differentiable at $x = 1$.
• At $x = 2$: There is a sharp corner at $x = 2$, so $g(x)$ is not differentiable at $x = 2$.
• At $x = 3$: The graph is smooth at $x = 3$, so $g(x)$ is differentiable at $x = 3$.
• At $x = 4$: Again, a sharp corner appears at $x = 4$, so $g(x)$ is not differentiable at $x = 4$.

Summary of Answers:

• $f$ continuous at $x = 1$: N
• $f$ continuous at $x = 2$: Y
• $f$ continuous at $x = 3$: Y
• $f$ continuous at $x = 4$: Y
• $f$ differentiable at $x = 1$: N
• $f$ differentiable at $x = 2$: N
• $f$ differentiable at $x = 3$: Y
• $f$ differentiable at $x = 4$: N
• $g$ differentiable at $x = 1$: Y
• $g$ differentiable at $x = 2$: N
• $g$ differentiable at $x = 3$: Y
• $g$ differentiable at $x = 4$: N

Let me know if you'd like further explanation on any specific concept!

Here are 5 relative questions:

1. What is the relationship between continuity and differentiability?
2. How do sharp corners affect the differentiability of a function?
3. Can a function be differentiable but not continuous? Why or why not?
4. How do we interpret open and closed circles on a graph in terms of function values?
5. What is the importance of the derivative in analyzing the smoothness of a curve?

Tip: If a function has a sharp corner or cusp, it means the left-hand and right-hand derivatives differ, making it non-differentiable at that point.