To sketch the graph of the function $f$ defined on $\mathbb{R}$ with a jump discontinuity at $x = 9$ and a removable discontinuity at $x = 7$, we can follow these steps:

Key Features:

1. Jump Discontinuity at $x = 9$:

   • At $x = 9$, the left-hand and right-hand limits are different, meaning there is a sudden "jump" in the graph.
   • The function will have two distinct values at $x = 9$, one on the left and one on the right.
   • The graph will have a break, with two separate points.

2. Removable Discontinuity at $x = 7$:

   • At $x = 7$, the function has a hole.
   • The left-hand and right-hand limits at $x = 7$ are equal, but the function is undefined at this point.
   • The graph will be continuous except for the missing point at $x = 7$.

Graph Description:

• For the sketch, the graph should be continuous elsewhere except at $x = 7$ and $x = 9$.
• Around $x = 9$, the graph should have a "jump" where the values on the left differ from the values on the right.
• At $x = 7$, the graph should have a hole (representing the removable discontinuity), but the graph should look smooth on either side of $x = 7$.

Would you like me to generate a visual sketch of this function, or would you like further details on any part?

---

Here are 5 related questions:

1. What is the difference between a jump and removable discontinuity?
2. How do you determine the limits of a function at a discontinuous point?
3. Can a function have more than one removable discontinuity?
4. How can you algebraically remove a removable discontinuity?
5. What is the impact of discontinuities on the derivative of a function?

Tip: A jump discontinuity indicates a break in the function's values, while a removable discontinuity is a hole that can be "filled" by redefining the function at the point.