For the first one you need a strong inequality sign with b (f(x) = x for a<=x<b); for the second, what happens if a = -1? Only a slight change is required..
Third is correct.
Also, if we let a and b be real number with a<b then what is an example of a bounded f on [a,b] for which sup_[a,b] f is not achieved?
What would be an example of a bounded, continuous f on [a, ) for which sup_[a, ) f is not achieved?
and what would be an example of a bounded, continuous f on [a,b) for which sup_[a,b) f is not achieved?
For the first one would it be : f(x)= { x, a<=x<=b
0, x=b}
second, f(x)= -1/x+1
third, f(x)=x ?