Say I have topological spaces $X$, $Y$ and $Z$, and a continuous map $f : X \times Y \to Z$. Say also that I have sequences $(x_i)$ in $X$ and $(y_i)$ in $Y$, along with points $x \in X$ and $y \in Y$ such that:
- $y_i \to y$ in $Y$, and
- $f(x_i, y_i) \to f(x,y)$ in $Z$.
What extra properties (if any) of the spaces $X, Y, Z$ or the map $f$ do I need in order to conclude that the sequence $(f(x_i, y))$ also converges to $f(x,y)$ in $Z$?
I am struggling to come up with any counterexamples, and I'm not sure whether that's because $f(x_i, y) \to f(x,y)$ quite generally, or if I'm just being unimaginative.
So far, I have managed to show that this convergence holds when $X,Y,Z$ are metric spaces (where product $X \times Y$ has product metric induced by an $L^p$-norm) and $f$ is Lipschitz continuous, as follows.
Take any $\epsilon > 0$. First, from $y_i \to y$ and Lipschitz continuity of $f$ with Lipschitz constant $K$ it follows that the sequence $f(\cdot,y_i)$ of functions $X \to Z$ converges uniformly to $f(\cdot,y)$: for there is some $N_1 \in \mathbb{N}$ such that for any $x' \in X$ and $i \geq N_1$, $$ d_Z( f(x',y_i) , f(x',y)) \leq K \left[ d_X (x',x')^p + d_Y(y_i,y)^p \right]^{1/p} = K d_Y(y_i,y) < \epsilon/2. $$
Also by convergence $f(x_i, y_i) \to f(x,y)$ there is some $N_2 \in \mathbb{N}$ such that $d_Z(f(x_i,y_i), f(x,y)) < \epsilon / 2$ for $i \geq N_2$.
Then by triangle inequality, $$ d_Z (f(x_i, y) , f(x,y)) \leq d_Z(f(x_i,y_i), f(x,y)) + d_Z(f(x_i,y_i), f(x_i,y)) < \epsilon, $$ for any $i \geq \operatorname{max}(N_1, N_2)$.
The case I am actually interested in has $X,Y,Z$ all non-compact smooth manifolds, and $f$ a smooth function. I could try to equip my manifolds with metrics and figure out if my map $f$ is Lipschitz continuous (e.g. by looking for a bound on the tangent map). But, I'm hoping that there's a sufficient condition of a more topological flavour that can guarantee $f(x_i, y) \to f(x,y)$.
