1. Consider the interval (0,1] in R. We need to show that there exists a continuous function f:(0,1] to R that is not bounded on (0,1]. A simple example of such a function is f(x) = 1/x. This function is continuous on the interval (0,1] and is not bounded as x approaches 0 from the right.

2. Now, we need to construct a continuous function f:[0, inf) to R which is not bounded. A simple example of such a function is f(x) = x. This function is continuous on the interval [0, inf) and is not bounded as x approaches infinity.

3. These examples illustrate the concept of compactness in the domain. A set in R is compact if it is closed and bounded. The interval (0,1] is a compact set because it is both closed and bounded. However, the function f(x) = 1/x is not bounded on this interval, which shows that a continuous function on a compact set is not necessarily bounded. On the other hand, the interval [0, inf) is not a compact set because it is not bounded. The function f(x) = x is also not bounded on this interval, which shows that a continuous function on a non-compact set can be unbounded.

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1. Consider the interval (0,1] in R. We need to show that there exists a continuous function f:(0,1] to R that is not bounded on (0,1]. A simple example of such a function is f(x) = 1/x. This function is continuous on the interval (0,1] and is not bounded as x approaches 0 from the right.

2. Now, we need to construct a continuous function f:[0, inf) to R which is not bounded. A simple example of such a function is f(x) = x. This function is continuous on the interval [0, inf) and is not bounded as x approaches infinity.

3. These examples illustrate the concept of compactness in the domain. A set in R is compact if it is closed and bounded. The interval (0,1] is a compact set because it is both closed and bounded. However, the function f(x) = 1/x is not bounded on this interval, which shows that a continuous function on a compact set is not necessarily bounded. On the other hand, the interval [0, inf) is not a compact set because it is not bounded. The function f(x) = x is also not bounded on this interval, which shows that a continuous function on a non-compact set can be unbounded.

Knowee AI · Accepted Answer

1. Consider the interval (0,1] in R. We need to show that there exists a continuous function f:(0,1] to R that is not bounded on (0,1]. A simple example of such a function is f(x) = 1/x. This function is continuous on the interval (0,1] and is not bounded as x approaches 0 from the right.

2. Now, we need to construct a continuous function f:[0, inf) to R which is not bounded. A simple example of such a function is f(x) = x. This function is continuous on the interval [0, inf) and is not bounded as x approaches infinity.

3. These examples illustrate the concept of compactness in the domain. A set in R is compact if it is closed and bounded. The interval (0,1] is a compact set because it is both closed and bounded. However, the function f(x) = 1/x is not bounded on this interval, which shows that a continuous function on a compact set is not necessarily bounded. On the other hand, the interval [0, inf) is not a compact set because it is not bounded. The function f(x) = x is also not bounded on this interval, which shows that a continuous function on a non-compact set can be unbounded.

consider the interval (0,1] element of R. show that there exist a continuous function f:(0,1] to R is not bounded on (0,1] construct a construct continuous function f:[0, inf) to R which is not bounded . what has got these examples to do with compactness of the domain

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