= Solution
Use $D=C([-1,1])$, restricted to $(-1,1)$ and regarded as a <linear subspace> of $L^p((-1,1))$. The <density of continuous functions in Lp on an interval> says that $D$ is a <dense subspace> whenever $1\leq p<\infty$.
For $g\in D$, the <supremum norm> gives
$$
|T_hg|\leq h^{-1/p'}\int_{-h}^h|g(x)|\,dx
\leq2\|g\|_\infty h^{1-1/p'}=2\|g\|_\infty h^{1/p}\longrightarrow0.
$$
In particular $D\subseteq Y$, with zero limit. Part (c), together with $D$ being a <dense subspace>, implies $Y=X$. To identify the limit throughout $X$, for $f\in X$ and $g\in D$ use
$$
|T_hf|\leq M\|f-g\|_p+2\|g\|_\infty h^{1/p},\qquad M=2^{1/p'}.
$$
First choose $g$ in the <dense subspace> to make the first term arbitrarily small, then let $h\to0^+$. This is <convergence on a dense subspace under a uniform operator bound>, and proves
$$
\boxed{Tf=\lim_{h\to0^+}T_hf=0\quad(f\in L^p((-1,1))),\qquad\|T\|=0.}
$$
The zero <linear functional> belongs to the <continuous dual space> $X^*$.
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