True or false, if $f \in C[D]$ and $D$ is closed, then $rng[f]$ is closed.
-False
-My first assumption would be to look at the Extreme Value Theorem:
-If $f \in C[D]$ where D is compact, then $\exists x^*,x_* \in D$ where $$F^* = f(x^*) = \begin{equation} \mathop{sup}_{\textbf{D}} \end{equation} (f)$$
$$F_* = f(x_*) = \begin{equation} \mathop{inf}_{\textbf{D}} \end{equation} (f)$$
-Since $f$ assumes the max and min points, then this implys that range of $f$ is closed.
-So, to disprove the initial statement, I can simply find an $f \in C[D]$ where $D$ is closed, but the range of $f$ is not. For example, let $$f(x) = e^{-x}, D = [0 , \infty)$$ The domain is closed, but the range of $f$ is $(0,1]$ but never assumes zero, so the infimum is not part of the set.
-Is this correct? I'm a little confused because the initial statement never suggests that the domain is bounded, thus it has the potential of not being compact. If this were the case, the EVT couldn't even be applied.
