Do we get a section on a ringed space $\mathcal{A}$ by mapping each element $x$ to the multiplicative unit element $1_x$ of $\mathcal{A}_x$?

Question

Suppose $(X, \mathcal{A})$ is a ringed space (where $\mathcal{A}$ is assumed to be a sheaf of unital, commutative rings over $X$) and consider the map \begin{eqnarray} \psi \colon X \to \mathcal{A} \qquad X \ni x \mapsto 1_x \in \mathcal{A}_x \end{eqnarray}

Is it true that this map must always be continuous, i.e. that $\psi$ is a section?

Answer 1

The element $1_x\in\mathcal{A}_x$ is just the image of the unit $1\in\mathcal{A}(X)$ under the restriction map $\mathcal{A}(X)\to\mathcal{A}_x$. So, your map is continuous, since it corresponds to the section $1\in\mathcal{A}(X)$.