Let $X_n$ be a continuous rv with cdf $F_n(x)$ and pdf $f_n(x)$. Let $X$ be a continuous rv with cdf $F(x)$ and pdf $f(x)$. It is known that convergence in distribution, i.e., $F_n(x)\to F(x)$ as $n\to\infty$ at every continuity point $x$ of $F(x)$, does not generally imply that $f_n(x)\to f(x)$ as $n\to\infty$. This question offers a counterexample. My question is: Is there a class of continuous rvs for which $F_n(x)\to F(x)$ does imply $f_n(x)\to f(x)$? What additional conditions are these rvs to satisfy? Incidentally, according to this question, pointwise convergence of $f_n(x)$ to $f(x)$ is enough to have $F_n(x)\to F(x)$.
I'm particularly interested in the case when $f_n(x)$ is supported on $[0,n]$ while $f(x)$ is supported on $[0,\infty)$. To boot, $f_n(x)$ is infinitely differentiable at every $x\in[0,n]$ for each $n$, and $f(x)$ is infinitely differentiable at every $x\ge0$. In fact, in my specific case, $f(x)=e^{-x}$, $x\ge0$.