In this lecture, we shall understand why differentiability implies continuity in complex analysis. This is a very important theorem. It tells us that if a complex function has a derivative at a point, then the function must be continuous at that point. So differentiability is stronger than continuity. But the reverse is not always true. A function may be continuous at a point and still fail to be complex differentiable there. This is the friendly way to remember it. Differentiability gives continuity, but continuity alone does not give differentiability.

Let $f$ be a complex function defined in a neighborhood of $z_0$. If $f$ is differentiable at $z_0$, then $f$ is continuous at $z_0$. In symbols, if $f'(z_0)$ exists, then $$\begin{equation*} \lim_{z\to z_0}f(z)=f(z_0). \quad \text{(1)} \end{equation*}$$