Does conclusion hold if convergence is pointwise?

Question

I just finished solving a proof that involved showing that if $f$ is continuous at $x$ and $f_n \to f$ uniformly then $x_n \to x$ in $X$ implies $f_n(x_n) \to f(x)$.

Now the question is, does this conclusion still hold if convergence is only point-wise? I am thinking it does not, but I'm having trouble coming up with a specific counter-example.

Answer 1

No, it's no longer true. Take the simplest $f$ possible -- the constant zero function.

Let $(f_n)_n$ defined by $$f_n(x) = \begin{cases} 1&\text{ if } x\in(0.\frac{1}{n})\\ 0&\text{ o.w.} \end{cases}$$ so that $f_n \xrightarrow[n\to\infty]{} f$ pointwise. But for $(x_n)_n$ defined by $x_n\stackrel{\rm def}{=}\frac{1}{2n}\xrightarrow[n\to\infty]{} x \stackrel{\rm def}{=}0$, we have $$\lvert f_n(x_n)-f(x)\rvert = f_n(x_n) = 1$$ for all $n\geq 1$.