Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions

Let $n$ be a natural number, let $E\subseteq\mathbb{R}^n$ be a subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, where for $s,t\in\mathbb{R}$ we write $s<t$ to mean that $s\le t$ and $s\ne t$, let $f:E\to\mathbb{R}$, and let $a\in E$.

Regard $\mathbb{R}^n$ as a metric space through the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and regard $\mathbb{R}$ as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line. In claim 1 the function $f$ is regarded as a map into $\mathbb{R}^m$ with $m=1$ and with $f$ as its single coordinate function.

Then the following hold.

1. (Agreement of the two notions) $f$ is continuous at $a$ in the Euclidean sense if and only if $f$ is continuous at $a$ relative to $E$, as a map from $E$ into the metric space $(\mathbb{R},d_{\mathbb{R}})$.

2. (Functions of class $C^2$ are semicontinuous) Let $U\subseteq\mathbb{R}^n$ be an open subset of $\mathbb{R}^n$ and let $u:U\to\mathbb{R}$ be of class $C^2$ on $U$. Then $u$ is continuous on $U$ as a map into $(\mathbb{R},d_{\mathbb{R}})$, and $u$ is both upper semicontinuous on $U$ and lower semicontinuous on $U$.