  The Behrend function of a $\mathbb C$-scheme $X$ is a constructible function
$\nu_X\colon X(\mathbb C) \to \mathbb Z$ introduced by Behrend, intrinsic to
the scheme structure of $X$. It is a (subtle) invariant of singularities of
$X$, playing a prominent role in enumerative geometry. To date, only a handful
of general properties of the Behrend function are known. In this paper, we
compute it for a large class of fat points (schemes supported at a single
point). We first observe that, if $X \hookrightarrow \mathbb A^N$ is a fat
point, $\nu_X$ is the sum of the multiplicities of the irreducible components
of the exceptional divisor $E_{X}\mathbb A^N$ in the blowup
$\textrm{Bl}_{X}\mathbb A^N$. Moreover, we prove that $\nu_X$ can be computed
explicitly through the normalisation of $\textrm{Bl}_{X}\mathbb A^N$.
  The proofs of our explicit formulas for the Behrend function of a fat point
in $\mathbb A^2$ rely heavily on toric geometry techniques. Along the way, we
find a formula for the number of irreducible components of $E_{X}\mathbb A^2$,
where $X \hookrightarrow \mathbb A^2$ is a fat point such that
$\textrm{Bl}_{X}\mathbb A^2$ is normal.
