The support of a distribution $T$ defined on an open set $\Omega$ is defined as follows : Let $\mathscr{A}$ be the set of open subsets $V \subset \Omega$ that satisfy : $\text{supp}\varphi \subset V \implies T(\varphi)=0$
Then the support of $T$ is the complement of $\displaystyle{ \cup_{V \in \mathscr{A}}} V$. Equivalently, for $x \in \Omega$, $x$ is in the support of $T$ iff for any neighbourhood $N$ of $x$ there exists $\varphi$ with support in $N$ such that $T(\varphi) \neq 0$.
What I mean by locally integrable distribution is a distribution $\varphi \mapsto \int \varphi f$ where $f$ is locally integrable. I would like an example proving that the support of $f$ can differ from that of its corresponding distribution.
An interesting comment by TZakrevskiy pointed out to me that the support of a locally integrable function may be a dodgy topic as it is only defined almost everywhere. However in my lecture notes it does precisely say that the support of $f$ may differ from thay of its distribution $T_f$.