In our early days of calculus, we said that a function is continuous if you could draw its graph without having to lift your pen. This is related to two concepts: that of the graph being closed, and that of it being connected. Indeed, a powerful theorem I've recently learned is that a linear function (and more generally, of any function if the range is compact) between Banach spaces has a closed graph in the product topology if and only if the function which corresponds to that graph is continuous.
Now, because every normed vector space is path connected, so must be its continuous image, and hence its graph, if I'm not mistaken. However, much less clear, is whether or not the converse holds:
If the graph of a function (either, linear between Banach spaces, or perhaps more generally) is path connected, does this imply that the function must have been continuous?
I see similar results for when the domain of the function has the property that bounded sequences contain convergent sub-sequences (example 1 and example 2), but that is of course not going to be true of the domain is an infinite dimensional normed vector space.
Thanks very much.