Consider a function $f(x,y): \mathbb{R}^2 \to \mathbb{R}$. I found the following phrase in a book: "$f(x,y)$ is continuous at $x_0$ uniformly in $y$". What does it mean?
I think that it means that $f(x,y) \to f(x_0,y)$ as $x \to x_0$ uniformly in $y$, that is for all $\varepsilon > 0$ there exists $\delta > 0$ such that $|f(x,y) - f(x_0, y)| < \varepsilon$ for all $(x,y)$ such that $|x - x_0| < \delta$ .
Could you please confirm if I am right and give some link to any book where it's written what does a phrase like "f(x,y) is continious at $x_0$ uniformly in $y$" mean?
I can't find such definition anywhere :(