Example of function $f:\mathbb{R}\to \mathbb{R}$ so that the sequence $f_n(x)=f(x+\frac{1}{n})$ not converge for all $x\in\mathbb{R.}.$

Question

I need and example of a function $f:\mathbb{R}\to \mathbb{R}$ so that the sequence $f_n(x)=f(x+\frac{1}{n})$ not converge for all $x\in\mathbb{R}$.

I have the following results:

1) If $f$ is continuous then $f_n\to f$ puntually.

2) If $f$ is continuous uniformly then $f_n\to f$ uniformly.

3) Exist a discontinuous function $f$ in $\mathbb{R}$ so that $f_n\to f$. (in this case my $f$ is the Dirichlet Function)

4) Exist a discontinuous function $f$ so that $f_n\to g$ with $g$ a continuous function. (in this case my $f$ is the Thomae Function)

5) If fix $a\in\mathbb{R}$, exist a function $f$ so that $f_n(a)$ not congerges.

I wait can you help me to find my example. Thanks!

Answer 1

For each residue class $A\in\mathbb{R}/\mathbb{Q}$ pick a representative $x_A,$ and define $f$ on $A$ by setting $f(x_A+p/q) = q,$ where $p$ and $q$ are relative prime and $q>0.$ We use the axiom of choice here.