I have started working on topology and there has been a question I can't figure out.
Consider a function $f$ defined on topological spaces $X\times Y$ with values in $\mathbb{R}$.
Both space are finite dimensional. X is endowed with euclidean topology (as a subspace of $\mathbb{R}^n$) and $Y$ is endowed with discrete topology (for example a discrete space $\{1,2,3,...,|Y|\}$). $f$ is linear on $X$.
By definition, we know that $f$ is continuous on $X$ as a linear application on a finite dimensional space. We also know that $f$ is continuous on $Y$ as $Y$ is endowed with discrete topology.
Therefore, we know that $f$ is continuous on $X$ and $f$ is continuous on $Y$. In general, I know that if a function is continuous on each variable it is not enough to deduce that $f$ is continuous on the product space of the variables.
However, I can't figure out if the function $f$ defined above is continuous on $X\times Y$.
Thank you for your help.
You have $$f^{-1}(O)=\{(x,y)\mid f(x,y)\in O\}=\bigcup_{y\in Y}\big\{(x,y)\mid f(x,y)\in O\big\}=\bigcup_{y\in Y}\Big(\big\{x\in X\mid f(x,y)\in O\big\}\times\{y\}\Big).$$ Note that you have an open set as the union of open sets on the right side.