Continuity/D to R/Point/Unit fractions/Power of tens/Exercise

Let ${}D\subseteq \mathbb {R}$ be a subset, ${}f\colon D\rightarrow \mathbb {R}$ a function and ${}a\in D$ a point. Show that the following properties are equivalent.

1. ${}f$ is continuous in ${}a$.
2. For every ${}n\in \mathbb {N} _{+}$, there exists some ${}m\in \mathbb {N} _{+}$ such that for

   ${}\vert {x-a}\vert \leq {\frac {1}{m}}\,$

   the estimate

   ${}\vert {f(x)-f(a)}\vert \leq {\frac {1}{n}}\,$

   holds.

3. For every ${}s\in \mathbb {N}$, there exists some ${}r\in \mathbb {N}$ such that for

   ${}\vert {x-a}\vert \leq {\frac {1}{10^{r}}}\,$

   the estimate

   ${}\vert {f(x)-f(a)}\vert \leq {\frac {1}{10^{s}}}\,$

   holds.

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