Countable real analysis

HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert’s proof of transcendence of the number $\mathrm{e}$. We also construct a uniformly continuous function $f:[0,1]\cap \mathbb{Q}\to \mathbb{R}$ such that $f^{\mathrm{\prime }}=1$ on $[0,1]\cap \mathbb{Q}$ and ${\lim }_{\begin{array}{c}a\to 1\mathrm{/}\sqrt{2}\\ a\in \mathbb{Q}\end{array}}f(a)=\frac{1}{\sqrt{2}}>f(b)$ for every $b\in [0,1]\cap \mathbb{Q}$.