  In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator
$\mathcal{B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{R})$
as $t\to 0$ provided $s\geq\frac{1}{4}$ -- more precisely -- on the one hand,
by constructing a counterexample in $\mathbb{R}$ we discover that the optimal
convergence index $s_{c,1}=\frac14$; on the other hand, we find that the
Hausdorff dimension of the disconvergence set for $\mathcal{B}_tf$ is
\begin{align*} \alpha_{1,\mathcal{B}}(s)&=\begin{cases} 1-2s&\ \ \text{as}\ \
\frac{1}{4}\leq s\leq\frac{1}{2};\\ 1 &\ \ \text{as}\ \ 0<s<\frac{1}{4}.
\end{cases} \end{align*} Moreover, Theorem 1.3 presents a higher dimensional
lift of Theorems 1.1- 1.2 under $f$ being radial.
