Prove a function is continuous (using partial derivative)?

Question

Given: $|f(x,y)_x'|<M, |f(x,y)_y'|<N near (0,0)$-partial derivative-

How may I prove that $f(x,y)$ is continuous in (0,0)?

I tried to prove that using $\epsilon$ but got to the end of the road since partial Derivative are not related. Any help?

Answer 1

Note that $|f(x,y)-f(0,0)| = |f(x,y)-f(x,0)+f(x,0)-f(0,0)| \le |f(x,y)-f(x,0)| + |f(x,0)-f(0,0)|$.

Now use the mean value theorem to bound $|f(x,y)-f(x,0)| $ and $ |f(x,0)-f(0,0)| $ separately.

$|f(x,y)-f(x,0)| \le N |x|$ and $|f(x,0)-f(0,0)| < M|y|$ so $|f(x,y)-f(0,0)| \le M|x|+N|y| \le \sqrt{N^2+M^2}\sqrt{x^2+y^2}$.

Now choose $\delta = {\epsilon \over \sqrt{N^2+M^2}}$.