  It is widely claimed that the quantile function is equivariant under
increasing transformations. We show by a counterexample that this is not true
(even for strictly increasing transformations). However, we show that the
quantile function is equivariant under left continuous increasing
transformations. We also provide an equivariance relation for continuous
decreasing transformations. In the case that the transformation is not
continuous, we show that while the transformed quantile at p can be arbitrarily
far from the quantile of the transformed at p (in terms of absolute
difference), the probability mass between the two is zero. We also show by an
example that weighted definition of the median is not equivariant under even
strictly increasing continuous transformations.
