Semicontinuity Under Negation and Characterization of Continuity

Let $(X,d)$ be a metric space, let $A\subseteq X$, let $\mathbb{R}$ be the set of real numbers with the addition and the order of its ordered field structure, let $u:A\to\mathbb{R}$, and let $x\in A$. Let $-u:A\to\mathbb{R}$ be the function whose value at $y\in A$ is the additive inverse of $u(y)$. Regard $\mathbb{R}$ as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line. Then the following hold.

1. (Negation) $u$ is lower semicontinuous at $x$ relative to $A$ if and only if $-u$ is upper semicontinuous at $x$ relative to $A$.

2. (Continuity) $u$ is continuous at $x$ relative to $A$, as a map from $A$ into the metric space $(\mathbb{R},d_{\mathbb{R}})$, if and only if $u$ is both upper semicontinuous at $x$ relative to $A$ and lower semicontinuous at $x$ relative to $A$.

Consequently, $u$ is continuous on $A$ if and only if $u$ is both upper semicontinuous on $A$ and lower semicontinuous on $A$.