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**Also sprach Zarathustra** Let $\displaystyle u=f(x,y)$ defined in $\displaystyle D$, continuous by $\displaystyle x$ and holds Lifchitz's condition for $\displaystyle y$, in other words: $\displaystyle |f(x,y_1)-f(x,y_2)|\leq A|y_1-y_2|$ where $\displaystyle (x,y_1),(x,y_2)$ are points in $\displaystyle D$ and $\displaystyle A$ is constant. Prove $\displaystyle f(x,y)$ continuous in $\displaystyle D$.

I know that $\displaystyle f(x,y)$ is continuous by $\displaystyle y$ (from Lifchitz's condition) and it's given that $\displaystyle f(x,y)$ is continuous by $\displaystyle x$, but now what?

Thank you!