Delta derivative of unilateral convolution

Theorem

If $f$ is $\Delta$-differentiable, then $$(f*g)^{\Delta}=f^{\Delta}*g+f(t_0)g,$$ and if $g$ is $\Delta$-differentiable, then $$(f*g)^{\Delta}=f*g^{\Delta}+fg(t_0),$$ where $(f*g)$ denotes unilateral convolution.