The graph provided shows a function $f(x)$ with discontinuities and possible points where the derivative does not exist. Let's analyze the graph to answer the questions:

(a) At what numbers $a$ does $\lim_{x \to a} f'(x)$ not exist?

To find where the derivative $f'(x)$ does not exist, we look for points on the graph where:

• The function has a sharp corner or cusp.
• The tangent line is vertical (which means the slope is undefined).

From the graph:

1. At $x = 1$, there is a sharp corner.
2. At $x = 3$, there is a vertical tangent.

Therefore, the derivative does not exist at $x = 1$ and $x = 3$.

Answer for (a):

$a = 1, 3$

(b) At what numbers $a$ is $f$ not continuous?

A function is not continuous at points where there are jumps, holes, or vertical asymptotes.

From the graph:

1. At $x = 2$, there is a jump discontinuity.

Answer for (b):

$a = 2$

(c) At what numbers $a$ does $\lim_{x \to a} f'(x)$ exist but $f$ is not continuous at $a$?

For this question, we need to find a point where the limit of the derivative exists but the function itself is not continuous. This is generally impossible because if a function is not continuous at a point, the derivative at that point also does not exist in the usual sense.

From the graph:

• There are no points where the limit of the derivative exists while the function is not continuous.

Answer for (c):

$a =$ (none)

Summary of Answers:

(a) $a = 1, 3$
(b) $a = 2$
(c) $a =$ (none)

Would you like a more detailed explanation of any part? Here are 5 related questions to explore further:

1. What are the conditions for a function to be differentiable at a point?
2. How can you determine if a limit exists using a graph?
3. What types of discontinuities are there, and how do they differ?
4. What is the relationship between continuity and differentiability?
5. How can we compute derivatives graphically without a formula?

Tip: To determine if a function is continuous at a point, check that the left-hand limit, right-hand limit, and the value of the function at that point are all equal.