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**1. Homework Statement**

Let [tex]f[/tex] be a real-valued function defined and continuous on the set of real numbers. Which of the following must be true of the set [tex]S={f(c):0<c<1}?[/tex]

I. S is a connected subset of the real numbers.

II. S is an open subset of the real numbers.

III. S is a bounded subset of the real numbers.

**2. Homework Equations**

none

**3. The Attempt at a Solution**

The answer is I and III only. I'm confused because I thought that I. implied II. Isn't a connected set necessarily open?