I have been stuck with the following problem for quite some time, as I will next describe, by first introducing the definitions at hand. Here I have two definitions of joint continuity and two definitions of separate continuity. It is supposed to be easy to prove, but I am having trouble proving claim 1 and claim 2. I would be grateful for any suggestions and happy to clarify any irregularities concerning the post
Joint Continuity
J1 A binary operation $\beta: R \times R \rightarrow R$ defined for all $x,y \in R$ by $\beta(x,y)=x * y$ is said to be jointly continuous at $(x,y) \in R \times R$ if and only if for each neighbourhood $W$ of $x * y$, there exists a product of open sets $U \times V \subseteq R \times R$ containing $(x,y)$ such that $U*V \subseteq W$. If $\beta$ is jointly continuous at each point in $R \times R$ then $\beta$ is called (globally) jointly continuous.
J2 Suppose that $X,Y$ and $Z$ are topological spaces. Let $f: X \times Y \rightarrow Z$ be a mapping. Then $f$ is called jointly continuous on $X \times Y$ if and only if the map $f$ is continuous from $X \times Y$, equipped with the product topology $\tau^{\times}$.
Claim 1 In the event that $X=Y=Z$, we have equivalence for J1 and J2.
Separate Continuity
S1 A binary operation $\beta: R \times R \rightarrow R$ defined for all $x,y \in R$ by $\beta(x,y)=x * y$ is said to be separately continuous if and only if for all $a \in R$ and for all $U \in \mathcal{N}_0$ there exists $V \in \mathcal{N}_0$ such that $a*V \subseteq U; V*a \subseteq U$
S2 Suppose $X,Y$ and $Z$ are topological spaces. Let $f: X \times Y \rightarrow Z$ be a mapping. Then $f$ is separately continuous on $X \times Y$ if and only if for all $(x_0,y_0) \in X \times Y$, both functions $x \mapsto f(x,y_0)$ and $y \mapsto f(x_0,y)$ are continuous on $X$ and $Y$ respectively.
Claim 2 In the event that $X=Y=Z$, we have equivalence for S1 and S2.