This question is within the context of Cauchy-Riemann equations. The text I am reading lists a theorem regarding the existence of the derivative of a function $f$ at point $z_0$. My question concerns a step in the proof of the theorem.
Let $f(z)=u(x,y)+iv(x,y) $
$\triangle{z}=\triangle{x}+i\triangle{y} $ ; $0<\lvert\triangle{z}\rvert<\epsilon$; $\triangle{w}=f(z_0+\triangle{z})-f(z_0)$
where $\triangle{u}=u(x_0+\triangle{x},y_0+\triangle{y})-u(x_0,y_0)$.
The proof states that if the first order partial derivative of $u$ is continuous at the point $(x_0,y_0)$, it follows that $\triangle{u}=u_x(x_0,y_0)\triangle{x}+u_y(x_0,y_0)\triangle{y}+\epsilon_1\triangle{x}+\epsilon_2\triangle{y}$.
Note: $\epsilon_1,\epsilon_2 \to 0$ as $(\triangle{x},\triangle{y}) \to (0,0) $
I do not understand how the previous result follows from the condition given. Any explanation would be greatly appreciated.