Let $A = \left[\frac{1}{x_i+y_j}\right]$, where $0<x_1<\cdots<x_n, 0<y_1<\cdots<y_n$ are positive real numbers. Then $\det(A)$ is a continuous function of variables $x_i, y_j$. Suppose $0<\lambda_1<\cdots<\lambda_n, 0<\mu_1<\cdots<\mu_n$ be positive reals such that $\lambda_{i+1}-\lambda_i = \alpha = \mu_{i+1}-\mu_i$ for all $1\leq i \leq n-1$, and $\alpha$ is a fixed positive integer.
It can be noted that the set $$ \{ (x_1,\ldots,x_n) \in (R^{+})^n: x_{i+1}-x_i = \alpha \quad \forall \quad 1\leq i \leq n-1 \}$$ is pathwise connected, so connected.
Let $$B = \left[\frac{1}{\lambda_i+\mu_j}\right].$$ Will $det(B)$ be continuous function of positive $\lambda_i, \mu_j$?
Any help will be appreciated. Thanks.
