> 
> Consider function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Suppose $\forall y \in \mathbb{R}$, the function $f(\cdot,y)$ is continuous and there is exists a positive number $K$ such that
> 
> 
> $$|f(x,y') - f(x,y'')| \leq K |y' - y''|, \forall x,y',y'' \in \mathbb{R}$$
> 
> 
> Show that $f$ is continuous on $\mathbb{R}^2$.
> 
> 
> 

I have no idea to solve this problem, so I need help. Thanks all!