Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable

Let $\mathbb{R}$ be the real numbers and let $|\cdot|$ be the absolute value on $\mathbb{R}$. We regard $\mathbb{R}$ both as the Euclidean space $\mathbb{R}^{1}$, carrying the Euclidean distance $d_{E}$, which is a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence, and as the metric space $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Numbers, in which $d_{\mathbb{R}}(s,t)=|s-t|$.

Let $f:\mathbb{R}\to\mathbb{R}$ and let $a\in\mathbb{R}$. We say that $f$ is Euclidean continuous at $a$ when it is continuous at $a$ as a map from $\mathbb{R}^{1}$ to $\mathbb{R}^{1}$, and that $f$ is metrically continuous at $a$ when it is continuous at $a$ relative to $\mathbb{R}$ as a map from $(\mathbb{R},d_{\mathbb{R}})$ to $(\mathbb{R},d_{\mathbb{R}})$.

Then the following hold.

1. (The two distances agree)(i) For all $s,t\in\mathbb{R}$, regarded as points of $\mathbb{R}^{1}$,

$$d_{E}(s,t)=|s-t|=d_{\mathbb{R}}(s,t).$$

2. (The two readings of continuity agree)(ii) $f$ is Euclidean continuous at $a$ if and only if $f$ is metrically continuous at $a$.

3. (A local Lipschitz bound suffices)(iii) Suppose there are $L,\rho\in\mathbb{R}$ with $0\le L$ and $0<\rho$ such that

$$|f(x)-f(a)|\le L\,|x-a|\qquad\text{for every }x\in\mathbb{R}\text{ with }|x-a|<\rho .$$

Then $f$ is Euclidean continuous at $a$, and metrically continuous at $a$.

4. (Sequential continuity)(iv) Suppose $f$ is Euclidean continuous at every point of $\mathbb{R}$. Then $f$ is sequentially continuous on $\mathbb{R}$ in the sense required by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable: whenever $(x_{k})_{k\in\mathbb{N}}$ is a sequence in $\mathbb{R}$ and $x\in\mathbb{R}$ are such that the sequence of Euclidean distances $(d_{E}(x_{k},x))_{k\in\mathbb{N}}$ converges to $0$, the sequence $(f(x_{k}))_{k\in\mathbb{N}}$ converges to $f(x)$.