Is this an open set in $(C([0,1]), \left\|{\cdot}\right\|_p)$?

Is this an open set in $(C([0,1]), \left\|{\cdot}\right\|_p)$?

Let $A=\{g\in C([0,1]):\int_{0}^{1}|g(x)|dx<1\}$. If $p\in [0,\infty]$, is
$A$ an open set of $(C([0,1]), \left\|{\cdot}\right\|_p)$?
Is it obvious that if $p=1$ then $A$ is open in $(C([0,1]),
\left\|{\cdot}\right\|_1)$, because $A=B(0,1)$.
I think $A$ is not open if $p>1$. Any hint to show this?
Thanks.