Let's analyze the function $f(x) = x$ defined on the interval $[0, 1]$.

1. Is the domain closed or not closed?

   • The domain $[0, 1]$ is a closed interval, as it includes its endpoints (0 and 1).

2. Is the domain bounded or unbounded?

   • The domain $[0, 1]$ is bounded because it is confined within the limits of 0 and 1.

3. Is the function bounded or unbounded?

   • The function $f(x) = x$ is bounded on the interval $[0, 1]$ because it takes values in the range [0, 1].

4. Is the function continuous (on its entire domain) or discontinuous (at least at one point of its domain)?

   • The function $f(x) = x$ is continuous on its entire domain $[0, 1]$. There are no points of discontinuity.

5. Does the function achieve an absolute maximum on its domain?

   • Yes, the function achieves an absolute maximum at $x = 1$, where $f(1) = 1$. It also achieves an absolute minimum at $x = 0$, where $f(0) = 0$.

Summary:

• Domain: Closed
• Domain: Bounded
• Function: Bounded
• Function: Continuous
• Absolute maximum: Achieved

Would you like more details or have any questions? Here are some related questions to consider:

1. What are the implications of the function being continuous on its domain?
2. How can you prove that $f(x) = x$ is continuous?
3. What other functions are bounded on the interval $[0, 1]$?
4. Can you find a function defined on $[0, 1]$ that is discontinuous?
5. How do you determine the absolute extrema of a function on a closed interval?

Tip: Understanding the properties of functions on closed intervals is crucial for calculus, especially in the context of the Extreme Value Theorem.