# Continuous image of a closed and bounded set in a metric space is closed?

I know that in metric space, continuous image of closed sets and continiuous image of bounded sets may not be closed or bounded. I am asking that if continuous image of a both closed and bounded set but not necessarily compact set is closed? Moreover, is it bounded? Thanks.

Take the metric space $$(\mathbb{R}^{+}, d_{\text{discr}})$$, that is, the positive real line with the discrete metric. The function

\begin{align*} f: (\mathbb{R}^+, d_{\text{discr}}) &\to (\mathbb{R}, |\cdot|) \\ x &\mapsto \frac1x \end{align*}

is continous, as a function whose domain is a discrete space. We have that $$f((0,1)) = (1,\infty).$$

The interval $$(0,1) \subseteq (\mathbb{R}^+, d_{\text{discr}})$$ is closed and bounded, while the interval $$(1,\infty) \subseteq (\mathbb{R}, |\cdot|)$$ is not closed and not bounded.