In the Above explanation it is written in square bracket that
Note that $x-({1\over 2})\mathrm dx$ and $x+({1\over 2})\mathrm dx$ are the values of X and in this interval f(x,y) may be treated as constant , I did not understand why and how f(x,y) can be treated as constant?
It can be treated as a constant because the range of $x$ is infinitesimally small. Since $f(x,y)$ (as a function of x for fixed y) is (presumably) smooth and bounded on this range, its variation goes to zero as the size of the range goes to zero and you can pull it out of the integral as a constant.