$\int_{\mathbb R} |f(x)| dx < \infty \implies \sum_{n\in \mathbb Z} |f(n)| < \infty $?

Question

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\infty$)

Let $f\in C_{0} (\mathbb R)$ such that $\int_{\mathbb R} |f(x)| dx < \infty $.

My Question: Can we expect $\sum_{n\in \mathbb Z} |f(n)| < \infty$; Or, we get a counter example, that is, $\sum_{n\in \mathbb Z} |f(n)| = \infty $ ?

Answer 1

Define $f$ to be $0$ everywhere except on intervals of the form $[n-1/n, n+1/n]$, $n$ an integer greater than $1$. On an interval $[n-1/n , n+1/n ]$ of this form, let $f$ be non-negative, continuous, and piecewise linear with maximum value $1/n$ at the midpoint of the interval and value $0$ at the endpoints.

Then $\int |f|\le\sum\limits_{n=1}^\infty 2/n^2<\infty$ and $\sum\limits_{n=1}^\infty|f(n)|=\sum\limits_{n=2}^\infty 1/n=\infty$.

Answer 2

This is a picture of David Mitra's example in the comment: