How can we show, rigorously, that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous map then its graph and the inverse of the graph (i.e the inverse of the relation) is a nowhere dense set? I'm trying to show that the plane cannot be covered by a countable number of graphs of continuous maps and its inverses, so by showing the above the result follows from the Baire category theorem.
Thanks in advance