For a surface S given explicitly as the graph of z = ƒ(x, y), where ƒ is a continuously differentiable function over a region R in the xy-plane, the surface integral of the continuous function G over S is given by the double integral over R. $$\iint_S G(x,y,z) dS=\iint_R G(x,y,z(x,y))\sqrt{1+f_x^2+f_y^2} dxdy$$
The definition requires that $f$ is continuously differentable, and $G$ is continuous. What are these requirement used for? What if I break the rules? Can anyone give an example or explain it directly? Thanks in advance : )
Imagine your surface $S$ is a sphere centered in the origin of coordinates, and you want to evaluate the surface $A$ of the upper part above the plane $z=0$, it is clear that $G=H(z)$ where $H$ is the Heaviside function. Although $H$ is discontinuous, your integral will still give the surface of the upper part of the sphere.\
The continuity of $f$ is required to define a normal vector given by the its gradient, but again, the normal vector coes not need to be continuous, i.e. if you have a corner you can always split the surface integral up into each of the sides and compute them as usual.