If $f(t)$ and $g(t)$ have bounded variation on $[0,1]$, then $\{(f(t),g(t)):t \in [0,1]\}$ cannot contain $[0,1]\times [0,1]$.
This is a problem I met in the analysis exam. Since $f(t)$ and $g(t)$ have bounded variation on $[0,1]$, we can see that $\{(f(t),g(t))\}$ is of bounded variation.