Question

1) Is it possible to have a function f defined on [ 2 , 4 ] and meets the given conditions?

Answer 1

The answer

1 and 2 are possibles, but 3 is not.

Answer 2

Answer: All the conditions 1, 2 and 3 are not possible.

Explanation:

A function is said to be continuous on [a,b] if it doesn't break on the interval [a,b].

1) If a function is f is defined on [2,4], it means for each input from [2,4] there exist a value of f. The function may or may not be continuous at [2,4]

If the function has minimum value f(4)=2, then the other values on the function must lies above this value and since f is continuous at [2,4), then a maximum value must be exist in the interval [2,4).

The given condition f is continuous on [ 2 , 4 ), minimum value f(4)=2, then it is not possible that the function has no maximum value.

2) If function is continuous on [4,5], it means it doesn't break from 4 to 5. It means the functions will take all the values lies between 4 to 5, either rational or irrational.

Therefore, If a function is f is defined on [4,5] and f is continuous on [ 4 , 5 ], then it is not possible that it will takes no rational values.

3) If the range of a function is in unbounded interval [a,b] then the function is not continuous on [a,b].

Therefore, If a function f defined on [ 2 , 5 ] and f is continuous on [ 2 ,5 ], then it is not possible that the range of f is an unbounded interval.