The set of all inner derivations $\mathcal{B}^1(A, X)$ is not closed in the set of continuous derivations $\mathcal{Z}^1(A, X)$,

Question

Let $A$ be a complex Banach algebra and $X$ be a Banach $A$-module

The set of all continuous $X$-derivations on $A$ is a complex linear space, denoted by $\mathcal{Z}^1(A, X)$. and The set of all inner $X$-derivations on $A$ is a complex linear subspace of $\mathcal{Z}^1(A, X)$, denoted by $\mathcal{B}^1(A, X)$

Provet that $\mathcal{B}^1(A, X)$ is not closed in The $\mathcal{Z}^1(A, X)$

Any comment or response is appreciated.