Thread: Limit and Continuity (multivariable calculus) [help]

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.
i.e. for every point (a,b) in the plane and for every epsilon>0, find delta>0 depending on a,b and epsilon such that whenever
0<|(x,y)-(a,b)|<delta, |f(x,y)-f(a,b)|<epsilon

How to find an epsilon in terms of delta? I am troubled in expressing epsilon as a function of delta.

Many thanks.

Originally Posted by catcat103

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.
i.e. for every point (a,b) in the plane and for every epsilon>0, find delta>0 depending on a,b and epsilon such that whenever
0<|(x,y)-(a,b)|<delta, |f(x,y)-f(a,b)|<epsilon

How to find an epsilon in terms of delta? I am troubled in expressing epsilon as a function of delta.

Many thanks.

epsilon is an arbitrary number.. you should find delta in terms of epsilon and not the other way around..

Originally Posted by catcat103

Question : For f(x,y)=xy, show that f(x,y) is continuous everywhere.

We shall prove is continous at . Let . Then . Similarly . Now use the inequality . Thus, . With these facts we can prove continuity. We need to show can be made arbitrary small. Add and subtact, if .
This tells us that for any choose .

ah...i forgot those useful inequalities. what a marvellous answer! thanks for your reply!