  The main result of this article is that we obtain an elementwise error bound
for the Fused Lasso estimator for any general convex loss function $\rho$. We
then focus on the special cases when either $\rho$ is the square loss function
(for mean regression) or is the quantile loss function (for quantile
regression) for which we derive new pointwise error bounds. Even though error
bounds for the usual Fused Lasso estimator and its quantile version have been
studied before; our bound appears to be new. This is because all previous works
bound a global loss function like the sum of squared error, or a sum of Huber
losses in the case of quantile regression in Padilla and Chatterjee (2021).
Clearly, element wise bounds are stronger than global loss error bounds as it
reveals how the loss behaves locally at each point. Our element wise error
bound also has a clean and explicit dependence on the tuning parameter
$\lambda$ which informs the user of a good choice of $\lambda$. In addition,
our bound is nonasymptotic with explicit constants and is able to recover
almost all the known results for Fused Lasso (both mean and quantile
regression) with additional improvements in some cases.
