Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

Question

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

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Here's a theorem from my real analysis book:

Assume $1\le p\le \infty$ and $f\in L^p(\mathbb{R}^n)$, $g\in L^{p'}(\mathbb{R}^n)$. Then the integral defining $(f*g)(x)$ exists for every $x\in\mathbb{R}^n$. The function $f*g$ thus defined is bounded and uniformly continuous on $\mathbb{R}^n$. Moreover, if $1 < p <\infty$, then also $f*g\in C_0$, meaning that $\lim_{|x|\to\infty}(f*g)(x)=0$.

(here $p$ and $p'$ are Holder conjugates, i.e., $1/p+1/p' = 1$)

Unfortunately this theorem doesn't take care the case $p=\infty$ for $f*g$ to be in $C_0$. Any idea for constructing $f$ and $g$? Thank you!

Answer 1

Let $g = 1$. Then $(f * g)(t) = \int f(x) g(t-x) dt = \int f$.

If $\int f \neq 0$, then $\lim_{t \to \infty} (f * g)(t) = \int f \neq 0$, of course.