Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem:
A map $f:X \rightarrow Y$ is continuous if and only if $f$ is connected in the product topology $X \times Y$.
Is this true? And if not, can anyone think of an additional premise or two that would make it true?