Sums and Nonnegative Multiples of Semicontinuous Functions

Let $(X,d)$ be a metric space, let $A\subseteq X$, let $\mathbb{R}$ be the set of real numbers with the addition and multiplication and the order $\le$ of its ordered field structure, let $u,v:A\to\mathbb{R}$, let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$, and let $x\in A$. Let $u+v:A\to\mathbb{R}$ and $\lambda u:A\to\mathbb{R}$ be the functions given by $(u+v)(y)=u(y)+v(y)$ and $(\lambda u)(y)=\lambda\,u(y)$ for $y\in A$. Then the following hold.

1. If $u$ and $v$ are upper semicontinuous at $x$ relative to $A$, then $u+v$ is upper semicontinuous at $x$ relative to $A$.

2. If $u$ is upper semicontinuous at $x$ relative to $A$, then $\lambda u$ is upper semicontinuous at $x$ relative to $A$.

3. If $u$ and $v$ are lower semicontinuous at $x$ relative to $A$, then $u+v$ is lower semicontinuous at $x$ relative to $A$; and if $u$ is lower semicontinuous at $x$ relative to $A$, then $\lambda u$ is lower semicontinuous at $x$ relative to $A$.

In particular, if the hypotheses of one of the three claims hold at every point of $A$, then the corresponding conclusion holds at every point of $A$.