Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem in Heat Kernels and Dirac operators makes use of the fact that in this situation $$\mathrm{ind}(D_0)=\mathrm{ind}(D_1).$$ The proof of the equality is discussed on page 125. However I feel like there is gap in the proof:

We seem so to assume that we can find a family $(D_t)_{t\in [0,1]}$ of Dirac operators such that the function $$[0,1]\ni t\mapsto\mathrm{ind}(D_t)\in\mathbb{R}$$ is continuous. Since the image is contained in $\mathbb{Z}$, it would be constant. In fact there is a very obvious candidate: As explained on page 117 we have that $A=D_1-D_0$ is function-linear, i.e. $A\in \Gamma(M,\mathrm{End}(E))$ and $$D_t=D_0+tA$$ is a Dirac operator for all $t\in[0,1]$. But how do we know that the above-mentioned function is continuous?

The authors suggest to use the McKean-Singer formula, but it was only proven for a Dirac operator which happens to be symmetric w.r.t. some metric. I assume that this is not the case for all Dirac operators (?), so can we avoid using McKean Singer? Or does the latter actually hold for all Dirac operators? Also the continuity would not be obvious to me even if we could use McKean Singer.