  Statistical functionals are called elicitable if there exists a loss or
scoring function under which the functional is the optimal point forecast in
expectation. While the mean and quantiles are elicitable, it has been shown in
Heinrich (2014) that the mode cannot be elicited if the true distribution can
follow any Lebesgue density. We strengthen this result substantially, showing
that the mode cannot be elicited if the true distribution is any distribution
with continuous Lebesgue density and unique local maximum. Likewise, the mode
fails to be identifiable relative to this class.
