A Hard thresholding operator $H_k:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is defined as a vector-valued function that maintains the top-k entries of a given vector in an absolute value sense and zero out the rest. As an example $H_2(x)=[-5,0,-3,0]^{\top}$ where $x=[-5,2,-3,1]^{\top}$ and $k=2$. (In a case where two entries are equal we keep the value with smallest index)
Question: Is $H_k$ a Lipschitz function that satisfies the following:
$$ ||H_k(x)-H_k(y)||_2 \leq c ||x-y||_2 \quad \quad x,y \in \mathbb{R}^n $$