  The space of $L^p$ graphons, symmetric measurable functions $w: [0,1]^2 \to
\mathbb{R}$ with finite $p$-norm, features heavily in the study of sparse graph
limit theory. We show that the triangular cut operator $M_{\chi}$ acting on
this space is not continuous with respect to the cut norm. This is achieved by
showing that as $n\to \infty$, the norm of the triangular truncation operator
$\mathcal{T}_n$ on symmetric matrices equipped with the cut norm grows to
infinity as well. Due to the density of symmetric matrices in the space of
$L^p$ graphons, the norm growth of $\mathcal{T}_n$ generalizes to the
unboundedness of $M_{\chi}$. We also show that the norm of $\mathcal{T}_n$
grows to infinity on symmetric matrices equipped with the operator norm.
