Convergence of a sequence of functions depending on a sequence of points Let the sequence $(x_k)_{k\in\mathbb{N}}$ converge to $x$, where $x_k\in D\subset \mathbb{R}$.
Let $(f_k)_{k\in\mathbb{N}}$ be a sequence of functions from $D$ to $\mathbb{R}$ such that
$
\lim_{k\to\infty} f_k(x) = f(x),
$ for any fixed $x\in D$.

How do you prove or disprove that
$$
\lim_{k\to\infty} f_k(x_k) = f(x).
$$
(I may guess it is necessary to assume $f(x)$ continuous)
Answer is negative.

Take $D=[0,1]$,$x_k=\frac{1}{k+1}$ and $f_k$ defined by:


$f_k(0)=0$
$f_k(x_k)=1$
$f_k(2 x_k)=0$
$f_k$ linear on $[0,x_k]$ and $[x_k,2 x_k]$
$f_k(x)=0$ for $x \in [2x_k,1]$


$f_k$ converges to the function equal to $0$ on $[0,1]$. However $f_k(x_k)=1$ for $k \in \mathbb{N}$ and $f(0)=0$.
No.

A counterexample, $D=[0,1]$, $f_n(x)=x^n$, then $f(x)=0, \forall x\in [0,1)$ and $f(1)=1$.

Pick $x_n=1-\frac{1}{n}, x=1$. $f(1)=1, f_n(x_n)= (1-\frac{1}{n})^n\to e^{-1}$