Why Are Open Domains Essential for Smooth Functions in Rn?

[jimisrv]

Hi, I have a quick question:

When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn?

Thanks!

 

[mathwonk]

ask yourself what it would mean to say the graph of a function defined on a one point set has a tangent line?

similarly what would it mean to say the graph of a function on a curve in the plane has a tangent plane?

smooth means there are tangent spaces to the graph, so the graph should be big enough to have a tangent plane.

so the domain has to be kind of fat, i.e. open. in order to have the right dimension.