Is a continuous function simply a connected function?
Solution 1:
It isn't true in general. An obvious variant of the Topologist's sine curve provides an example of a function $f:\Bbb R\rightarrow \Bbb R$ whose graph is connected but fails to be continuous (at $x=0$).
However, this article shows that "it is correct to conclude that continuous real functions over $\Bbb R$ are those functions over $\Bbb R$ whose graphs, in the plane $\Bbb R^2$, are both closed and connected".