The question you're presenting involves a mathematical setup with a function $f$ from a metric space $(X, \rho)$ to a metric space $(U, \sigma)$, and a subset $D$ in $X$.

Let me break it down based on the two parts of the question:

(a) First part:

If $f(D)$ is bounded, then $f$ is bounded, and the goal is to show that $\text{diam}(A) = \text{diam}(\overline{A})$.

Here, $A$ is likely a subset of $D$ (or could be any arbitrary set within $D$). The diameter of a set $A$, denoted $\text{diam}(A)$, is the supremum of the distances between any two points in $A$.

(b) Second part:

If $f$ is continuous on $D$ and is homeomorphic (i.e., it has a continuous inverse) on $D$, then we are asked to prove that $f$ is continuous on $X$.

Let me clarify your request: would you like a more detailed explanation or solution steps to any part of the problem, or perhaps a more specific focus on one of the questions?