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## Homework Statement

Prove

## f(x,y,z)=xyw## is continuos using the Lipschitz condition

## Homework Equations

the Lipschitz condition states:

##|f(x,y,z)-f(x_0,y_0,z_0)| \leq C ||(x,y,z)-(x_0,y_0,z_0)||##

with ##0 \leq C##

## The Attempt at a Solution

##|xyz-x_0y_0z_0|=|xyz-x_0y_0z_0+x_0yz-x_0yz|\leq|(x-x_0)(yz)|+|x_0(yz-y_0z_0)|##

##\leq|(x-x_0)(yz)|+|x_0(yz-y_0z_0+yz_0-yz_0)| \leq |yz(x-x_0)|+|x_0[y(z-z_0)+z_0(y-y_0)]|##

## \leq |yz(x-x_0)|+|x_0y(z-z_0)|+|x_0z_0(y-y_0)|##

and by choosing ##C=max\{|yz|,|x_0y|,|x_0z_0|\}## I have my inequality.

I'm not sure I can do this, though. Are all the passages logic?

Thank you in advance :)