Diffusion Equation with a noncontinuous Auxiliary Condition

Question

I am currently trying to show that if we are given the Diffusion Equation $u_t=cu_{xx}$ with Auxiliary Condition $u(x,0)=g(x)$ where $g$ is non continuous at a point $x_0$, we get the statement: $$\lim_{t\rightarrow 0^+}u(x_0,t)=\frac{1}{2}[g(x_0+)+g(x_0-)]$$ Where $g(x_0+)=\lim_{\tau\rightarrow x_0^+} g(\tau)$, and an analogous statement holds for $g(x_0-)$. I proved this fine when $g$ is continuous, but I don't know where to begin with this one. We denote $u(x,t)$ as: $$u(x,t) = \int_{\infty}^\infty \Phi(x-y)g(y) dy$$ Where $\Phi$ is the Heat Kernel. Now I proved it analytically (the case where $g$ continuous), but if someone could show me how to relate the noncontinuous case to the continuous case I would appreciate it. The wya I approached the continuous case was showing that $\forall\epsilon$ $\exists\delta$ such that if $0<t<\delta$ then we had: $$\left|\int_{-\infty}^\infty \Phi(x-y)g(y) dy\right|<\epsilon$$ The proof was involved, and thus I am not going to restate it here, and also I'm not expecting a full length answer for this, I just want to know where the similarities lie. Thank you

Answer 1

First show that the result holds for the function $$ H(x)=\begin{cases}0 & \text{if }x<0,\\1 & \text{if }x\ge0.\end{cases} $$ Then consider the function $f(x)=g(x)+c\,H(x-x_0)$, where the constant $c$ is chosen so that $f$ is continuous at $x_0$.