Is the convolution pointwise bounded?

Question

A problem from an old exam:

Prove or disprove: if $p,q \in [1,\infty)$ such that $p^{-1}+q^{-1}=1$ and $f\in L^p, g\in L^q$, then the convolution $f*g$ is pointwise bounded.

First of all: what did they possibly mean by pointwise boundedness? (As far as I know it's a property of a sequence of functions but $f*g$ is a standalone function).

I know that if $r^{-1}=p^{-1}+q^{-1}-1$ then $\|f*g\|_r \leq \|f\|_p \|g\|_q$. Can it be used here?

Answer 1

$$ |(f*g)(x)=\int f(x-y)g(y)\, dy| \leq \left( \int |f(x-y)|^p \, dy \right)^{1/p} \left( \int |g(y)|^q \, dy \right)^{1/q} = \|f\|_p \|g\|_q $$ for (almost) all $x$. Hence $\|f*g\|_\infty \leq$...