Are functions converging towards the delta distribution at $x$ necessarily positive at $x$?

Question

Consider a sequence of non-negative functions $(f_k)_{k\in\mathbb{N}}\subset C(\mathbb{R}^d)$ which converges (in the distributional sense) towards the delta-distribution $\delta_x$ at some $x\in\mathbb{R}^d$. I was wondering whether or not this implies that $f_k(x)>0$ for almost every $k\in\mathbb{N}$?

Answer 1

It doesn't. Consider hat functions of height $2\epsilon$ between 0 and $\epsilon$