What are same general properties of having f(x,x) for joint density f(x,y)?

Question

For two random variables X, Y with joint density f(x,y). What are the properties of f(x,x) and can we bound $$\int_{-\infty}^{\infty}f(x,x) dx \quad and \quad \int_{-\infty}^{\infty}\frac{\partial f(x,x)}{\partial y} dx?$$ And can we normalise the f(x,x) to be a density function?