Question

1) Is it possible to have a function f defined on [ 2 , 4 ] and meets the given conditions?
f is continuous on [ 2 , 4 ), minimum value f(4)=2, and no maximum value.

2) Is it possible to have a function f defined on [ 4 , 5 ] and meets the given conditions?
f is continuous on [ 4 , 5 ], takes on no rational values.

3) Is it possible to have a function f defined on [ 2 , 5 ] and meets the given conditions?
f is continuous on [ 2 ,5 ] and the range of f is an unbounded interval.

Answer 1

i am thinking of a counter example for the first one, but i can't seem to come up with one
i bet you can come up with and example for #2, think of a constant function

Answer 2

#3 contradicts whatever theorem it is that says a continuous function on a closed interval has a max and a min

Answer 3

i bet @Zarkon can come up with an example for the first one

Answer 4

So far we have that for 2 it is possible, but for 3 it isn't possible?

Answer 5

ooh i got one

Answer 6

who said art was dead?

Answer 7

that makes a lot of sense!

Answer 8

so 1 is possible, 2 is possible, but 3 isn't possible. correct?

Answer 9

i was thinking of \[\frac{1}{4-x},2\] or something similar