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We investigate the completeness and completions of the normed algebras$D^{(1)}(X)$ for perfect, compact plane sets $X$. In particular, we construct aradially self-absorbing, compact plane set $X$ such that the normed algebra$D^{(1)}(X)$ is not complete. This solves a question of Bland and Feinstein. Wealso prove that there are several classes of connected, compact plane sets $X$for which the completeness of $D^{(1)}(X)$ is equivalent to the pointwiseregularity of $X$. For example, this is true for all rectifiably connected,polynomially convex, compact plane sets with empty interior, for allstar-shaped, compact plane sets, and for all Jordan arcs in $\mathbb{C}$.  In an earlier paper of Bland and Feinstein, the notion of an$\mathcal{F}$-derivative of a function was introduced, where $\mathcal{F}$ is asuitable set of rectifiable paths, and with it a new family of Banach algebras$D_{\mathcal{F}}^{(1)}(X)$ corresponding to the normed algebras $D^{(1)}(X)$.In the present paper, we obtain stronger results concerning the questions when$D^{(1)}(X)$ and $D_{\mathcal{F}}^{(1)}(X)$ are equal, and when the former isdense in the latter. In particular, we show that equality holds whenever $X$ is'$\mathcal{F}$-regular'.  An example of Bishop shows that the completion of $D^{(1)}(X)$ need not besemisimple. We show that the completion of $D^{(1)}(X)$ is semisimple wheneverthe union of all the rectifiable Jordan arcs in $X$ is dense in $X$.  We prove that the character space of $D^{(1)}(X)$ is equal to $X$ for allperfect, compact plane sets $X$, whether or not $D^{(1)}(X)$ is complete. Inparticular, characters on the normed algebras $D^{(1)}(X)$ are automaticallycontinuous.

Normed algebras of differentiable functions on compact plane sets