# Is it sufficient to have a complex partial derivative for the complex partial derivative be continous?

Suppose we have $$\phi(z,w)$$ a function of two complex variables and that for each fixed $$w$$ the function $$z\mapsto \phi(z,w)$$ is holomorphic, that is, it exists $$\frac{\partial}{\partial z}\phi(z,w)$$. Is $$\frac{\partial}{\partial z}\phi(z,w)$$ continous? I know that by Osgood lemma that if both $$\frac{\partial}{\partial z}\phi(z,w)$$ and $$\frac{\partial}{\partial w}\phi(z,w)$$ exist then the answer is yes, but if only one of them exist?

It depends on what you mean by $$\frac{\partial f}{\partial z}$$ existing, since there are really two interpretations of that. A common one, when applied to smooth functions, is the Wirtinger derivative, but that's probably not what you mean. If you mean that the limits $$\lim_{h \to 0} \frac{f(z+h,w)-f(z,w)}{h} \qquad \text{and} \qquad \lim_{h \to 0} \frac{f(z,w+h)-f(z,w)}{h}$$ exist at all $$(z,w)$$, then yes, this means that the function is holomorphic and holomorphic functions are in fact analytic, that is, given by a power series in several variables at all points. The derivatives are also analytic (and hence also holomorphic) and so definitely continuous.