The winding number of a closed curve undergoing continuous deformation There is a proof of the fundamental theorem of algebra that uses the fact that the winding number around the origin of a closed curve undergoing continuous deformation remains constant unless the curve passes through the origin. Usually the explanation given for why the winding number remains constant is that the winding number (which is restricted to integer values) is a continuous function of the curve. Is there another way to explain why the winding number remains constant?

If you deform a given curve slightly, say by pushing a little finger out, the difference in the integral of $1/z$ will be an integral $\int_C 1/z\,dz $ around a closed curve $C$ not containing the pole at the origin. Thus the integral will be zero by the residue theorem. The proof is completed by noting that any deformation between two curves can be achieved by a sequence of small deformations.