I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused.
Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in L^p(\Bbb R)$ and $ g \in L^q(\Bbb R)$ then $f*g$ is uniformly continuous and $f*g(x) \to 0 $ as $x \to \infty$ and $x \to -\infty$, where "$*$" denotes convolution.