I am asked to show that the $2-$dimensional Lebesgue measure of the graph of a continuous real function is zero.
Could you give me some hints how I could show it??
Consider the restriction to a compact interval $[a,b]$. Use the uniform continuity of $f$ on the compact interval to show that you can cover that part of the graph by open rectangles of total measure $< \varepsilon$, for any $\varepsilon > 0$.