Sketch the graph of a function f that is defined on double-struck R and continuous except for the stated discontinuities.
jump discontinuity at 9, removable discontinuity at 7

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? 

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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.

Sketch the graph of a function f that is defined on double-struck R and continuous except for the stated discontinuities: jump discontinuity at 9, removable discontinuity at 7.

Learn how to sketch the graph of a function with a jump discontinuity at x=9 and a removable discontinuity at x=7. This problem explores key concepts in real analysis and graphing functions, suitable for students in Grades 11-12.

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