Let me first analyze the graph in the image you uploaded so that I can answer the questions regarding the function $F(x)$.

I'll evaluate the limits and the continuity at the specified points based on the graph. Just a moment!I analyzed the graph in the image you uploaded and will now answer the questions one by one:

a) $\lim_{x \to 1} F(x)$

Looking at the graph as $x$ approaches 1 from both sides:

• From the left side ($x \to 1^-$), the value of the function is 1.
• From the right side ($x \to 1^+$), the value is also 1. Thus, the limit exists and $\lim_{x \to 1} F(x) = 1$.

b) Is $F(x)$ continuous at $x = 1$?

From the graph, there is no jump or discontinuity at $x = 1$, and the function is defined at this point. So, the answer is yes, the function is continuous at $x = 1$.

c) $\lim_{x \to 2} F(x)$

Looking at the graph as $x$ approaches 2:

• From the left side ($x \to 2^-$), the function value is 2.
• From the right side ($x \to 2^+$), the value is also 2. So, $\lim_{x \to 2} F(x) = 2$.

d) Is $F(x)$ continuous at $x = 2$?

At $x = 2$, there is a removable discontinuity because the function has a jump at this point. Thus, $F(x)$ is not continuous at $x = 2$.

e) $\lim_{x \to 3} F(x)$

Looking at the graph as $x$ approaches 3:

• From the left side ($x \to 3^-$), the function value is 3.
• From the right side ($x \to 3^+$), the value is also 3. Therefore, $\lim_{x \to 3} F(x) = 3$.

f) Is $F(x)$ continuous at $x = 3$?

At $x = 3$, there is no discontinuity, and the function is well-defined at this point. Therefore, the function is continuous at $x = 3$.

Would you like further details on any of these points or have other questions related to limits and continuity? Here are 5 related questions to expand on this topic:

1. What is the difference between a jump discontinuity and a removable discontinuity?
2. How can you determine if a function is continuous from a graph?
3. What is the significance of one-sided limits in determining continuity?
4. How does the limit of a function behave near vertical asymptotes?
5. Can a function have a limit at a point but not be continuous there?

Tip: When analyzing limits, always check from both sides (left and right) to ensure they match for the limit to exist.