Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev space.

Let $\psi\in C_c^\infty(M)\subseteq H^0(M)$. Then in particular $(D+t)^{-1}\psi$ is smooth. Also, for any $\phi\in C_c^\infty(M)$, the value of the function $(D+t)\phi$ at $x\in M$ is simply

$$((D+t)\phi)(x) = D\phi(x) + t\phi(x),$$

which in particular depends continuously on the parameter $t$. I'm interested in whether this is true for the inverse operator. More precisely:

**Question:** At a fixed point $x\in M$, does the value $((D+t)^{-1}\psi)(x)$ depend continuously on $t$?