If $1\le p,q\le \infty$, $\frac{1}{p}+\frac{1}{q}=1$, $f\in L^p(\mathbb{R}^n), g\in L^q(\mathbb{R}^n)$, I would like to show that $f*g$ is bounded and continuous in $\mathbb{R}^n$.
I'm able to prove that it is bounded by Hölder's inequality: \begin{aligned} \left|\int f(x-y)g(y)\,dy\right| &\le \int \left|f(x-y)g(y)\right|dy \\ &\le \|f(x-\cdot)\|_{L^p} \| \|g\|_{L^q} \\ &\le \|f\|_{L^p} \|g\|_{L^q}. \end{aligned} However, I'm not sure how to show continuity. I thought of using the DCT but $f,g$ aren't necessarily bounded.
EDIT: Responding to a comment below, I compute \begin{aligned} |f*g(x_1)-f*g(x_2)| &\le \int |g(y)||f(x_1-y)-f(x_2-y)|\,dy \\ &\le \|g\|_{L^q} \| f(\cdot)-f(\cdot+x_2-x_1)\|_{L^p} \\ &=\|g\|_{L^q} \| f(\cdot)-f(\cdot+h)\|_{L^p} \end{aligned} where we take $h=x_1-x_2\to 0$. So need to show that $f(x+h)\to f(x)$ in $L^p$ as $h\to 0$.
Here are the details for $1\le p < \infty$. Let $h\to 0$ and \begin{align*} |\int [f(x+h-y)-f(x-y)]g(y)\,dy| \le \|f(x+h-\cdot)-f(x-\cdot)\|_{L^p(dy)}\|g\|_q. \end{align*} By translation and reflection invariance of Lebesgue integration, $\|f(x+h-y)-f(x-y)\|_{L^p(dy)} = \|f(y-h)-f(y)\|_{L^p(dy)}$. Now you can show $\|f(y-h)-f(y)\|_{L^p(dy)}\to 0$ in an elementary fashion by first proving it for the case $f = \chi_E$ is a step function, and then for simple functions and ultimately general $L^p$ functions using the monotone convergence theorem.
Another, somewhat less elementary, approach would be to use the density of smooth and compactly supported functions in $L^p$ and use the mean value theorem, or the continuous compactly supported functions and the bounded/dominated convergence theorem... There are many possibilities for how to establish the continuity in $L^p$.
For $p = \infty$, you can recycle the same argument by putting the $x$ and $x+h$ into the argument of $g$ instead of $f$ using that convolution is symmetric.