[proofplan] We prove both directions. The forward direction takes the [limit](/page/Limit) defining $f'(w)$ along the real and imaginary axes separately, extracting $f'(w) = u_x + iv_x$ and $f'(w) = v_y - iu_y$. Equating real and imaginary parts yields the Cauchy--Riemann equations. Real [differentiability](/page/Derivative) follows because multiplication by $f'(w)$ is a real-linear map. The reverse direction uses real differentiability and the Cauchy--Riemann equations to factor the increment as $(u_x + iv_x) \cdot h + o(|h|)$, showing this is complex-linear. [/proofplan] [step:Forward direction: restrict the complex limit to the real and imaginary axes] Suppose $f'(w)$ exists. Taking the [limit](/page/Limit) $z \to w$ along the real axis ($z = w + h$, $h \in \mathbb{R}$, $h \to 0$): \begin{align*} f'(w) = \lim_{h \to 0} \frac{f(w + h) - f(w)}{h} = \lim_{h \to 0} \frac{u(c+h, d) - u(c,d)}{h} + i\lim_{h \to 0} \frac{v(c+h, d) - v(c,d)}{h} = u_x + iv_x. \end{align*} Taking the limit along the imaginary axis ($z = w + ik$, $k \in \mathbb{R}$, $k \to 0$): \begin{align*} f'(w) = \lim_{k \to 0} \frac{f(w + ik) - f(w)}{ik} = \frac{u_y}{i} + v_y = v_y - iu_y. \end{align*} Equating real and imaginary parts of $u_x + iv_x = v_y - iu_y$ gives the Cauchy--Riemann equations: $u_x = v_y$ and $v_x = -u_y$. [/step] [step:Forward direction: deduce real differentiability from complex differentiability] The existence of $f'(w)$ means $f(w + h) = f(w) + f'(w) \cdot h + o(|h|)$ as $h \to 0$ in $\mathbb{C}$. Writing $f'(w) = \alpha + i\beta$ and $h = h_1 + ih_2$, the multiplication $f'(w) \cdot h = (\alpha h_1 - \beta h_2) + i(\beta h_1 + \alpha h_2)$ is the [linear map](/page/Linear%20Map) $(h_1, h_2) \mapsto (\alpha h_1 - \beta h_2, \beta h_1 + \alpha h_2)$, represented by the Jacobian matrix \begin{align*} \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix} = \begin{pmatrix} \alpha & -\beta \\ \beta & \alpha \end{pmatrix}. \end{align*} This is precisely real [differentiability](/page/Derivative) of $(u, v)$ at $(c, d)$. [/step] [step:Reverse direction: reconstruct complex differentiability from real differentiability and the Cauchy--Riemann equations] Suppose $u$ and $v$ are real-differentiable at $(c, d)$ and the Cauchy--Riemann equations $u_x = v_y$, $u_y = -v_x$ hold. Real differentiability gives: \begin{align*} u(c + h_1, d + h_2) &= u(c,d) + u_x h_1 + u_y h_2 + o(|(h_1, h_2)|), \\ v(c + h_1, d + h_2) &= v(c,d) + v_x h_1 + v_y h_2 + o(|(h_1, h_2)|). \end{align*} Setting $h = h_1 + ih_2$ and using $u_y = -v_x$, $v_y = u_x$: \begin{align*} f(w + h) - f(w) &= (u_x h_1 + u_y h_2) + i(v_x h_1 + v_y h_2) + o(|h|) \\ &= (u_x h_1 - v_x h_2) + i(v_x h_1 + u_x h_2) + o(|h|) \\ &= (u_x + iv_x)(h_1 + ih_2) + o(|h|) \\ &= (u_x + iv_x) \cdot h + o(|h|). \end{align*} Dividing by $h$ and taking $|h| \to 0$ gives $f'(w) = u_x + iv_x$. [guided] The crucial step in the reverse direction is factoring the linear terms as complex multiplication. Real differentiability produces four terms: $u_x h_1$, $u_y h_2$, $v_x h_1$, $v_y h_2$. The Cauchy--Riemann equations $u_y = -v_x$ and $v_y = u_x$ allow substitution. The real part of the increment becomes $u_x h_1 + u_y h_2 = u_x h_1 - v_x h_2$. The imaginary part becomes $v_x h_1 + v_y h_2 = v_x h_1 + u_x h_2$. Now observe that $(u_x + iv_x)(h_1 + ih_2) = (u_x h_1 - v_x h_2) + i(v_x h_1 + u_x h_2)$, which is exactly the expression above. So $f(w+h) - f(w) = (u_x + iv_x) \cdot h + o(|h|)$. This factorisation works because the Jacobian matrix has the special form $\begin{pmatrix} \alpha & -\beta \\ \beta & \alpha \end{pmatrix}$ (with $\alpha = u_x$, $\beta = v_x$), which is precisely the matrix of multiplication by the complex number $\alpha + i\beta$. Without the Cauchy--Riemann equations, the Jacobian would be a general $2 \times 2$ matrix, and no complex number could reproduce it. The technical note $o(|(h_1, h_2)|) = o(|h|)$ uses $|(h_1, h_2)| = \sqrt{h_1^2 + h_2^2} = |h_1 + ih_2| = |h|$, so the real and complex error terms are identical. [/guided] [/step]