Given pointwise convergence of $f_n \rightarrow f$, does $\lim_{\epsilon \rightarrow 0} f(x+\epsilon) = f(x)$ implies uniform convergence?

Question

I was wondering if provided that the partial sums $f_n$ converge to $f$ matching pointwise convergence, what this would imply:

$$ \lim_{\varepsilon \rightarrow 0} f(x+\varepsilon) = f(x) $$

If that would be the case, for all $x$, wouln't this imply uniform convergence? And if not, why.

Many thanks in advance!

Answer 1

Example: let $f_n(x)= \frac{nx}{1+n^2x^2}$. Then $ (f_n)$ converges pointwise to the function $f(x)=0$ on $ \mathbb R.$ Then we have $\lim_{\varepsilon \rightarrow 0} f(x+\varepsilon) = f(x)$ for all $x$. But $(f_n)$ does not converge uniformly, since $f_n(1/n)=1/2$ for all $n$.