Differentiation under the sign of double integral

Consider the double integral $$I(a)=\int_0^1\int_0^1 f(x,y,a)dxdy$$What are the conditions needed so that we can differentiate under the integral sign of this double integral? That is when do we have $$I'(a)=\int_0^1\int_0^1 \frac{\partial}{\partial a} f(x,y,a)dxdy$$Is the condition that $f(x,y,a)$ is continuous and its partial derivative $\frac{\partial}{\partial a} f(x,y,a)$ is continuous on $x,y\in(0,1)$ enough?

Any reference will be highly appreciated. Thank you.