Theorem: Suppose that $$f(z)=u(x,y)+i\,v(x,y)$$ and for $z_0 = x_0 + y_0i$, the first order partial derivatives of $u$ and $v$ all exist and are continuous at a point $(x_0,y_0)$. Then if $u$ and $v$ satisfy the Cauchy-Riemann equations, $f$ is differentiable at $z_0$
I read the proof of this from the complex analysis book (https://math.unice.fr/~nivoche/pdf/Brown-Churchill-Complex%20Variables%20and%20Application%208th%20edition.pdf), and it details writing $\Delta f = \Delta u + \Delta vi$, we use the continuity of the partial derivatives to write $\Delta u = u(x_0 + \Delta x, y_0 + \Delta y)$ as:
$$\Delta u = u_x(x_0,y_0)\Delta x + u_y(x_0,y_0)\Delta y + \epsilon_1 \Delta x + \epsilon_2\Delta y$$
where $\epsilon_1$ and $\epsilon_2$ go to $0$ and $\Delta x$ and $\Delta y$ go to $0$. We do this similarly for $\Delta v$. The rest of the proof doesn't rely on the continuity of the partial derivatives, only that $\epsilon_1$ and $\epsilon_2$ go to $0$. It seems however that we can write $\epsilon_1$ and $\epsilon_2$ as:
$$\epsilon_1 = \frac{u(x_0 + \Delta x,y_0) - u(x_0,y_0)}{\Delta x} - u_x(x_0,y_0)$$ $$\epsilon_2 = \frac{u(x_0 + \Delta x,y_0 + \Delta y) - u(x_0 + \Delta x,y_0)}{\Delta y} - u_y(x_0,y_0)$$
We can make these piecewise, so that they equal $0$ for when $\Delta x$ and $\Delta y$ are respectively $0$. Because the all of the partial derivatives of $y$ exist in a neighborhood of $(x_0,y_0)$, we use the mean value theorem for $g(y) = u(x_0 + \Delta x, y)$ on the interval $[y_0, y_0 + \Delta y]$ to write the equation on the left of $\epsilon_2$ as $u_y(x_0 + \Delta x, y*)$ for some $y^*$. Since $y^*$ is bounded by $y_0 + \Delta y$ it tends to $y_0$ as $\Delta y$ goes to $0$. It seems we can then use continuity of $u_y$ to get that this goes to $u_y(x_0,y_0)$, so that $\epsilon_2$ goes to $0$.
Because all we need is for these to tend to $0$ as $\Delta x$ and $\Delta y$ go to 0, I'm confused since it seems we don't need continuity to establish that $\epsilon_1$ goes to $0$, because the equation on the left in $\epsilon_1$ is just the definition of the partial derivative at $(x_0,y_0)$. Therefore, to establish the $\Delta u$ equation, it seems all we need then is that at least one of the partial derivatives is continuous for $u$ and that the partial derivatives all exist at $(x_0,y_0)$.
I read another proof at this link (https://personal.math.ubc.ca/~feldman/m300/cauchyRiemann.pdf) which goes into more detail surrounding the equations, but it seems like, using similar equations to these, we also don't need continuity at both of the partial derivatives to establish the relevant equation in this either. Could someone explain to me where I'm going wrong?
