Can anyone suggest a mathematical analysis textbook that contains proof of this Proposition below?
I have already sought proof of this result in classic textbooks of mathematical analysis such as Rudin and Zorich, but without success.
Background. Let $I^{n}\overset{_\mathrm{def}}{=}\left\{x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}|\;\;a_i\leq x_{i} \leq b_{i}\,;\; i=1, \ldots, n\right\}$ be an $n$-dimensional closed interval and $I$ a closed interval $[a, b] \subset \mathbb{R}$.
Proposition. Let $f \in C^0(I^n\times I,\mathbb{R})$. Then the function $$ m(x_1,\ldots,x_{n-1},x_{n},y)=\min_{t\in [y,b] } f(x_1,\ldots,x_{n-1},x_{n},t) $$ is also continuous on the $I^n\times > I$.
Here the number $m(x_1,\ldots,x_{n-1},x_{n},y)$ means that for any fixed point $(x_1,\ldots,x_{n-1},x_{n})\in I^n$ we have $ m(x_1,\ldots,x_{n-1},x_{n},y) \leq f(x_1,\ldots,x_{n-1},x_{n},t) $ for all $t\in [y,b]$.
