Can we construct a nonzero function $f(x)$ near (but not at a point) such that the integral evaluates to $1$?

Question

I want to construct a function that is nonzero between $[-t,0) \cup (0,t]$, as $t \to 0$, such that the limit of the integral across $[-t,0) \cup (0,t]$ equals $1$. The function can be scaled in amplitude by $t$.

Also outside of $t$, the function must equals zero.