Closed unbounded operator with domain not closed

One of the simplest examples is the following:

Let $D(T) = C^1[0,1]$ be the space of of continuously differentiable functions (one-sided derivative at the end points) and let $X = C^0[0,1]$ be equipped with the norm $\lVert f \rVert_\infty = \sup_{x \in [0,1]} \lvert f(x)\rvert$.

Then the derivative $T = \frac d{dx}$ is an operator $D(T) \subset C^0[0,1] \to C^0[0,1]$ and $T$ is easily checked to have closed graph. Remember: if $f_n \to f$ pointwise and $f_{n}' \to g$ uniformly then $g$ is the derivative of $f$ by an application of the fundamental theorem of calculus.

However, $D(T)$ is not closed (it is dense but not all of $C^0[0,1]$) and $T$ is not bounded (consider $x^n$).