Proof Detail: Cauchy-Riemann Equations Imply Holomorphy

Question

I'm working through the proof of Theorem 2.4 in Chapter 1 of Stein/Shakarchi's Complex Analysis. I'm looking for clarification on where exactly we use the hypothesis that $u$ and $v$ are continuously differentiable. My guess is that it's used right away when writing $$u(x + h_1, y + h_2) - u(x,y) = \frac{\partial u}{\partial x}h_1 + \frac{\partial u}{\partial y}h_2 + |h|\psi(h)$$ and similarly for $v$, since that equation comes from $u$ being differentiable as a map $\mathbb{R}^2 \to \mathbb{R}$, which we can only say provided we know that the partials of $u$ are continuously differentiable. Is this correct?