= Solution
Let $C_n$ be the union of the $2^n$ closed intervals remaining after $n$ steps of the <Cantor set> construction. Their total length is $(2/3)^n$. Any point outside $C_n$ has a ternary expansion with a $1$ among its first $n$ digits, so $A^c\subseteq C_n$. Endpoints of deleted intervals also belong to $A$ because they have an alternative ternary expansion containing a $1$, as in $2/3=0.1222\ldots{}_3$.
A partition using all endpoints of $C_n$ has lower Darboux sum at least
$$
1-\left(\frac23\right)^n.
$$
Since $A$ is dense, every upper Darboux sum is one. Letting $n\to\infty$ gives equal upper and lower integrals, and therefore
$$
\boxed{f\text{ is Riemann integrable and }\int_0^1f(x)\,dx=1}.
$$
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