If $\phi:X\times Y\rightarrow Z$ is continuous then is $\varphi(x_1,x_2,y):=\phi(x_1,y)-\phi(x_2,y)$ continuous too

Question

So let be $X$ and $Y$ topological spaces and let be $Z$ a topological vector space. So if $\phi:X\times Y\rightarrow Z$ is a continuous function then we define a function $\varphi:X\times X\times Y\rightarrow Z$ thorugh the equation
$$
\varphi(x_1,x_2,y):=\phi(x_1,y)-\phi(x_2,y)
$$
for any $(x_1,x_2,y)\in X\times X\times Y$ and thus let we prove that it is continuous. So we observe that the statement follows directely proving that the function $\Delta:X\times X\times Y\rightarrow Z\times Z$ defined through the equation
$$
\Delta(x_1,x_2,y):=\big(\phi(x_1,y),\phi(x_2,y)\big)
$$
for any $(x_1,x_2,y)\in X\times X\times Y$ is continuous since $\varphi$ would be composition of continuous functions. So to prove hte continuity of $\Delta$ I tried to implement the universal mapping theorem for products but unfortunately I did no able to conclude anything. So could someone help me, please?

Answer 1

The function $\Delta : (x_1,x_2,y) \mapsto (\phi(x_1,y),\phi(x_2,y))$ is continuous iff the two functions $f:(x_1,x_2,y) \mapsto \phi(x_1,y)$ and $g:(x_1,x_2,y) \mapsto \phi(x_2,y)$ are continuous.

To see that this is the case, notice that $f$ is the composition of $\phi$ and $(x_1,x_2,y) \mapsto (x_1,y)$ and that both are continuous.