I wish to show that the map F: R^3 -> R^2 given by F(x,y,z)=(0,5*((e^x)+y),0,5*((e^x)-y)) is continuous.

I can argue that F is continuous because it consists of functions G(x,y)=0,5*((e^x)+y) and H(x,y)=0,5*((e^x)-y) which themselves are continuous.

But how can I show that the map is continuous by using the definition of continuity which involves the preimage?

(sorry but the latex-function is not working)

Appreciate the help.