I don't know the context, but that sure looks like a consequence of u being a step function - and I've often seen them written with the name "u".

If u is the unit step function at 0 (i.e. u(s) = 1 for s>=0, and u(s) = 0 for s<0), then your formula drops out, since then h(t,x)=a(x) when x<t, and h(t,x)=0 when x>t.