= Solution
For a <continuous function> $g$ on $[0,1]$, the <L-infinity norm> of its restriction to $(0,1)$ equals its <supremum norm> on $[0,1]$. Indeed, values arbitrarily close to a maximum occur on an interval of positive <Lebesgue measure>, even if the maximum lies at an endpoint. Also, <continuous functions> equal <almost everywhere> on $(0,1)$ are equal everywhere on $[0,1]$. Thus restriction embeds $C([0,1])$ isometrically as a <linear subspace> of $X$.
The <point evaluation functional> $E_{x_0}(g)=g(x_0)$ is complex-linear and satisfies $|E_{x_0}(g)|\leq\|g\|_\infty$. The constant <function> one attains equality, so $\|E_{x_0}\|=1$. The <complex Hahn-Banach theorem> extends this <bounded linear functional> from that <linear subspace> to $X$, preserving its <operator norm>. Therefore
$$
\boxed{\ell_{x_0}\in X^*,\qquad\|\ell_{x_0}\|=1,\qquad\ell_{x_0}(g)=g(x_0)\quad(g\in C([0,1])).}
$$
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