the Pearson correlation is an important measure to detect linear association between two random variables, but is it continuous function?
I'm trying see whether the following is true: $\lim_{C \to c} \rho(X, Z + CY) = \rho(X, Z + cY)$.
where $\rho$ is the Pearson correlation and C, c are scalars. It seems trivial for me but I couldn't prove it