thanks I think I got it :)
What you need is that if A is connected and f is continuous, then f(A) is connected, since the only connected sets in R are the intervals. "Compact" and bounded won't help since there is no requirement here that the interval be bounded. And knowing something is a subset of [a, b] doesn't tell you anything about that "something" being an interval!
I think you need to use the "intermediate value" property of continous functions: If y is any number between f(a) and f(b), then there exist c in [a, b] such that f(c)= y.
Separate names with a comma.