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The problem states:

Let \( g \) be a continuous function on the closed interval \([0, 1]\), with \( g(0) = 1 \) and \( g(1) = 0 \). The task is to identify which of the following statements is **NOT necessarily true**:

(A) For all \( a \) and \( b \) in \([0, 1]\), if \( a = b \), then \( g(a) = g(b) \).  
(B) There exists a number \( h \) in \([0, 1]\) such that \( g(h) = \frac{1}{2} \).  
(C) There exists a number \( h \) in \([0, 1]\) such that \( g(h) = \frac{3}{2} \).  
(D) For all \( h \) in the open interval \( (0, 1) \),  
\[
\lim_{x \to h} g(x) = g(h).
\]

### Analysis of the Options:
- **(A)**: This statement is always true because \( g \) is continuous, and by definition of continuity, if \( a = b \), then \( g(a) = g(b) \). Thus, this statement is **true**.
  
- **(B)**: By the Intermediate Value Theorem, since \( g(0) = 1 \) and \( g(1) = 0 \), there must be some \( h \in [0, 1] \) where \( g(h) = \frac{1}{2} \). So, this statement is **true**.

- **(C)**: This statement is **false** because \( g(1) = 0 \) and \( g(0) = 1 \), and the values of \( g \) in the interval \([0, 1]\) are constrained between 0 and 1 due to the continuity of \( g \). Therefore, there cannot be any \( h \) in \([0, 1]\) such that \( g(h) = \frac{3}{2} \), which is greater than 1.

- **(D)**: This statement is a standard result from the definition of continuity, which holds for any continuous function on an interval. Thus, it is **true**.

### Conclusion:
The statement that is **NOT necessarily true** is **(C)**, since the function cannot take values greater than 1 in the interval \([0, 1]\).

Would you like further details or explanations?

Here are five related questions:
1. What is the Intermediate Value Theorem and how is it applied here?
2. Why does continuity imply that \( g \) cannot take values larger than 1 in this context?
3. How does the limit definition in option (D) relate to the concept of continuity?
4. Can you provide an example of a continuous function that satisfies the conditions in the problem?
5. What would change if \( g(0) \) and \( g(1) \) were different values?

**Tip:** Continuity ensures that a function cannot "jump" in value within an interval—it must pass through all intermediate values.

Let g be a continuous function on the closed interval [0, 1]. Let g(0) = 1 and g(1) = 0. Which of the following is NOT necessarily true? (A) For all a and b in [0, 1], if a = b, then g(a) = g(b). (B) There exists a number h in [0, 1] such that g(h) = 1/2. (C) There exists a number h in [0, 1] such that g(h) = 3/2. (D) For all h in the open interval (0, 1), lim g(x) = g(h).

Explore a problem involving a continuous function on [0, 1] with given boundary conditions, applying the Intermediate Value Theorem and continuity. Suitable for college-level calculus students.

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