The sensitivity in differential privacy with deep learning

In differentially private deep learning, the sensitivity is determined by clipping gradient norm (see Abadi et al.'s paper). In this paper, when the clipping gradient norm is $$C$$, the sensitivity is $$C$$. Why is the sensitivity $$C$$? I think the sensitivity should be $$2C$$.

Definition 1.  A randomized mechanism $$\mathcal M: \mathcal D\to\mathcal R$$ with domain $$\mathcal D$$ and range $$\mathcal R$$ satisfies $$(\epsilon,\delta)$$-differential privacy if for any two adjacent inputs $$d,d'\in\mathcal D$$ and for any subset of outputs $$S\subseteq\mathcal R$$ it holds that $$\Pr[\mathcal M(d)\in S]≤e^\epsilon\Pr[\mathcal M(d')\in S]+\delta.$$
Since each record contributes a value in $$[-C,C]$$ to the gradient, removing or adding a record will change the total by $$C$$. It would be $$2C$$ if the definition included "arbitrarily changing a record".