Translation operator and continuity

First of all, in the proof you need to assume something else about the function $g$. Usually one takes $g\in C_c(\mathbb{R}^n)$, the space of continuous functions with compact support. This implies that $g$ is uniformly continuous, allowing one to prove that $\|g(x-a)-g(x)\|_p\to0$ as $a\to0$.

The proof shows that $$ \lim_{a\to0}\int_{\mathbb{R}^n}|f(x-a)-f(x)|^p\,dx=0. $$ This is very different of continuity of $f$ , which would be $$ \lim_{a\to0}|f(x-a)-f(x)|=0\quad\forall x\in\mathbb{R}^n. $$

Your statement in the comments that an operator $T$ is continuous on a Banach space $X$ if, for a sequence $f_n \rightarrow f$ in $X$ then we have $Tf_n \rightarrow Tf$. That is about the operator $T$ being continuous. The functions $f_n$ and $f$ are simply an arbitrary collection of elements in the space that form a convergent sequence and its limit.

Consider that the definition of a continuous operator makes sense even on a space where the elements are not functions themselves!