In this wiki link, it gives the definition of densely defined operator. After the definition, it gives an example of a densely defined unbounded operator, differentiation operator on $C^0[0,1]$. First, the differentiation is well-defined in $C^1[0,1]$
My question is why "This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of $C^0([0, 1]; R)$.'' What problem would we have when trying to extend the differentiation operator?