The graph of a continuous function $\mathbb{R}\to \mathbb{R}$ is Lebesgue null in ${\mathbb{R}}^2$

Fix a real $\eta >0$. For each integer $m$, the interval $[m,m+1]$ is compact by [F3], so [L3] gives a real ${\delta }_m>0$ such that $x,y\in [m,m+1]$ and $\mathrm{\mid }x-y\mathrm{\mid }<{\delta }_m$ imply $\mathrm{\mid }f(x)-f(y)\mathrm{\mid }<\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-3}$. By [F4] choose a natural number $N_m\ge 1$ with $1\mathrm{/}{N}_m<{\delta }_m$, and put $t_j:=m+j\mathrm{/}{N}_m$ for $0\le j\le N_m$. Then for every $j$ with $1\le j\le N_m$ and every $x\in [t_{j-1},t_j]$ one has $\mathrm{\mid }x-t_{j-1}\mathrm{\mid }\le 1\mathrm{/}{N}_m<{\delta }_m$, hence $\mathrm{\mid }f(x)-f(t_{j-1})\mathrm{\mid }<\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-3}$.

For each such subinterval, the corresponding graph piece lies in the closed box $[t_{j-1},t_j]\times [f(t_{j-1})-\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-3},f(t_{j-1})+\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-3}]$, whose width is $t_j-t_{j-1}$ and whose height is $\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-2}$. Therefore [L2] gives a box cover of ${\mathrm{\Gamma }}_f\cap ([m,m+1]\times \mathbb{R})$ whose total area is ${\sum }_{j=1}^{N_m}(t_j-t_{j-1})\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-2}=\eta 2^{-\mathrm{\mid }m\mathrm{\mid }-2}$.

Summing these covers over all integers $m$ and using [F1], the whole graph is covered by countably many closed boxes with total area at most $\eta {\sum }_{m\in \mathbb{Z}}{2}^{-\mathrm{\mid }m\mathrm{\mid }-2}=\eta (2^{-2}+2{\sum }_{n=1}^{\mathrm{\infty }}{2}^{-n-2})=3\eta \mathrm{/}4<\eta$, the geometric-series identity coming from [F5]. Since $\eta >0$ was arbitrary, [L1] gives that ${\mathrm{\Gamma }}_f$ is Lebesgue null.