= Solution
Take $K>0$. For $t>0$, the diffusion <fundamental solution of a linear differential operator> $F$ satisfies $(\partial_t-K\partial_x^2)F=0$ and has initial value $\delta(x)$ in the sense of <distributions>:
$$
\boxed{\lim_{t\downarrow0}\int_\mathbb R F(x,t)\psi(x)\,dx=\psi(0)}
$$
for every <smooth> compactly supported <test function> $\psi$. With the usual spatial decay it has unit mass. Equivalently the causal extension $\Theta(t)F(x,t)$ is a <retarded fundamental solution> satisfying $(\partial_t-K\partial_x^2)(\Theta F)=\delta(x)\delta(t)$.
The causal <Green function> is zero for $t<\tau$ and satisfies
$$
\boxed{(\partial_t-K\partial_x^2)G(x,t;y,\tau)=\delta(x-y)\delta(t-\tau)}.
$$
For $t>\tau$ it solves the homogeneous <diffusion equation>, and its initial jump is $G(x,\tau^+;y,\tau)=\delta(x-y)$, again distributionally. The natural spatial decay and a standard bounded or integrable solution class specify the usual kernel, avoiding unrestricted growing solutions. The PDE acts on $x,t$; $y,\tau$ label the location and time of the unit impulse. Values of the <Heaviside step function> exactly at the jump do not affect these <distribution> identities.
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