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The question discusses the Intermediate Value Theorem (IVT) and asks for two sketches based on a function \( f \) defined on the interval \([1, 4]\), with some discontinuity at \( x = 3 \). Here's the analysis:

### Intermediate Value Theorem (IVT):
The IVT states that if a function is continuous on a closed interval \([a, b]\), and \(N\) is any value between \(f(a)\) and \(f(b)\), then there exists at least one point \(c \in (a, b)\) such that \(f(c) = N\).

In this case:
- \(f(1) = 1\)
- \(f(4) = 3\)
- There is a discontinuity at \(x = 3\).
- You are given \(N = 2\) and need to explore whether the function \(f\) might or might not satisfy the conclusion of the IVT, given that it's discontinuous at \(x = 3\).

### Two scenarios:
1. **Graph where the function does not satisfy the conclusion of the IVT:**
   - The function could "jump" at \(x = 3\), meaning there's a break or gap at that point.
   - Since the function is discontinuous at \(x = 3\), it's possible that the function never attains the value \(f(x) = 2\) in the interval \([1, 4]\).
   - In this case, the discontinuity prevents the IVT from applying.

2. **Graph where the function might still satisfy the conclusion of the IVT:**
   - Even though there is a discontinuity at \(x = 3\), the function could still take the value \(2\) at some point between \(1\) and \(4\), perhaps at \(x = 2\).
   - The function could be continuous on \([1, 3)\) and \((3, 4]\), with a discontinuity at \(x = 3\), but still pass through the value \(2\) at some point, which would satisfy the conclusion of the IVT.

Would you like me to create the sketches for both cases or explain the details further?

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### Related Questions:
1. What are the key conditions for the Intermediate Value Theorem to apply?
2. Why is the discontinuity at \(x = 3\) significant for the IVT?
3. Could there be other values for \(N\) that the function \(f\) must satisfy under the IVT?
4. What happens to the IVT if a function is only piecewise continuous?
5. How does the behavior of \(f(x)\) at \(x = 3\) affect the possibility of satisfying the IVT?

### Tip:
When verifying the IVT, always check for both the continuity condition and the value condition in the interval.

Suppose that a function f is continuous on [1, 4] except at x=3 (therefore it does not satisfy the hypotheses of the Intermediate Value Theorem), and that f(1)=1 and f(4)=3. Let N=2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem, and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem.

Explore how the Intermediate Value Theorem applies to a function f(x) that is continuous on [1, 4] except at x=3. Learn how to sketch two graphs, one showing that f might not satisfy the IVT and one showing that f might still satisfy the IVT conclusion.

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