Let
f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0 < c < 1} ?
I. S is a connected subset of R.
II. S is an open subset of R.
III. S is a bounded subset of R.
(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III
Answer key:
SPOILER: C
I'm confused by the answer key suggesting that S is a connected subset of R.
As {c: 0 < c < 1} is an open subset of R and f(c) is continuous, I think {f(c): 0 < c < 1} should be an open subset of R also. Isn't it??