The energy dispersion relation of graphene with the tight-binding approximation and interactions up to nearest neighbors is $$E(k_1,k_2)=\pm |t|\sqrt{3+2\cos(ak_1) + 4\cos\left(\frac{a}{2}k_1\right)\cos\left(\frac{\sqrt{3}a}{2}k_2\right)}.$$ I am trying to show that around $K:=\frac{2\pi}{3a}(1,\sqrt{3})$, the energy relation is linear, that is, up to first order $E(K+v)\propto \|v\|.$ The problem I'm having is that I can't do a Taylor expansion around $K$ since the function is not differentiable at $K$. Any help would be appreciated!
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You expect $E(K+v)$ to be proportional to $\lVert v\rVert$ which is not differentiable at $0$. Then why should $E$ be differentiable at $K$? In any case, you can Taylor expand everything under the root and get the result you need.