  We prove that there exists a countable family of continuous real functions
whose graphs together with their inverses cover an uncountable square, i.e. a
set of the form $X\times X$, where $X$ is an uncountable subset of the real
line. This extends Sierpi\'nski's theorem from 1919, saying that $S\times S$
can be covered by countably many graphs of functions and inverses of functions
if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also
motivated by Shelah's study of planar Borel sets without perfect rectangles.
